9 28

9_27

9_29

Visit 9 28's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 28's page at Knotilus! Visit 9 28's page at the original Knot Atlas! 9 28 Quick Notes

9 28 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3849 X11,15,12,14 X5,13,6,12 X13,7,14,6 X15,18,16,1 X9,16,10,17 X17,10,18,11 X7283
Gauss code	-1, 9, -2, 1, -4, 5, -9, 2, -7, 8, -3, 4, -5, 3, -6, 7, -8, 6
Dowker-Thistlethwaite code	4 8 12 2 16 14 6 18 10
Conway Notation	[21,21,2+]

Minimum Braid Representative: Length is 9, width is 4. Braid index is 4.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index $\{4,6\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-9][-2] Hyperbolic Volume 11.5632 A-Polynomial See Data:9 28/A-polynomial

[edit Notes for 9 28's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $3$ Rasmussen s-Invariant -2

[edit Notes for 9 28's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-5t^{2}+12t-15+12t^{-1}-5t^{-2}+t^{-3}$
Conway polynomial	$z^{6}+z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 51, -2 }
Jones polynomial	$-q^{2}+3q-5+8q^{-1}-8q^{-2}+9q^{-3}-8q^{-4}+5q^{-5}-3q^{-6}+q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{6}+a^{6}-2z^{4}a^{4}-5z^{2}a^{4}-4a^{4}+z^{6}a^{2}+4z^{4}a^{2}+7z^{2}a^{2}+5a^{2}-z^{4}-2z^{2}-1$
Kauffman polynomial (db, data sources)	$z^{4}a^{8}-z^{2}a^{8}+3z^{5}a^{7}-4z^{3}a^{7}+2za^{7}+4z^{6}a^{6}-4z^{4}a^{6}+2z^{2}a^{6}-a^{6}+3z^{7}a^{5}+2z^{5}a^{5}-9z^{3}a^{5}+6za^{5}+z^{8}a^{4}+8z^{6}a^{4}-17z^{4}a^{4}+12z^{2}a^{4}-4a^{4}+6z^{7}a^{3}-5z^{5}a^{3}-7z^{3}a^{3}+6za^{3}+z^{8}a^{2}+7z^{6}a^{2}-19z^{4}a^{2}+14z^{2}a^{2}-5a^{2}+3z^{7}a-3z^{5}a-4z^{3}a+3za+3z^{6}-7z^{4}+5z^{2}-1+z^{5}a^{-1}-2z^{3}a^{-1}+za^{-1}$
The A2 invariant	$q^{22}-q^{18}+q^{16}-3q^{14}-q^{12}+4q^{6}+3q^{2}-q^{-2}+q^{-4}-q^{-6}$
The G2 invariant	$q^{114}-2q^{112}+4q^{110}-6q^{108}+4q^{106}-3q^{104}-4q^{102}+13q^{100}-20q^{98}+27q^{96}-25q^{94}+15q^{92}+6q^{90}-29q^{88}+54q^{86}-62q^{84}+53q^{82}-29q^{80}-10q^{78}+49q^{76}-72q^{74}+73q^{72}-44q^{70}+3q^{68}+32q^{66}-53q^{64}+37q^{62}-7q^{60}-33q^{58}+56q^{56}-58q^{54}+23q^{52}+30q^{50}-81q^{48}+105q^{46}-99q^{44}+53q^{42}+8q^{40}-67q^{38}+103q^{36}-103q^{34}+79q^{32}-24q^{30}-28q^{28}+63q^{26}-64q^{24}+43q^{22}-34q^{18}+52q^{16}-35q^{14}+5q^{12}+41q^{10}-70q^{8}+77q^{6}-52q^{4}+5q^{2}+39-70q^{-2}+76q^{-4}-56q^{-6}+24q^{-8}+8q^{-10}-33q^{-12}+39q^{-14}-32q^{-16}+19q^{-18}-5q^{-20}-5q^{-22}+7q^{-24}-8q^{-26}+5q^{-28}-2q^{-30}+q^{-32}$

Further Quantum Invariants

Further quantum knot invariants for 9_28.

A1 Invariants.

Weight	Invariant
1	$q^{15}-2q^{13}+2q^{11}-3q^{9}+q^{7}+q^{5}+3q-2q^{-1}+2q^{-3}-q^{-5}$
2	$q^{42}-2q^{40}-q^{38}+6q^{36}-6q^{34}-4q^{32}+15q^{30}-8q^{28}-11q^{26}+18q^{24}-3q^{22}-13q^{20}+8q^{18}+4q^{16}-6q^{14}-5q^{12}+9q^{10}+3q^{8}-14q^{6}+9q^{4}+12q^{2}-16+3q^{-2}+13q^{-4}-10q^{-6}-3q^{-8}+7q^{-10}-2q^{-12}-2q^{-14}+q^{-16}$
3	$q^{81}-2q^{79}-q^{77}+3q^{75}+3q^{73}-6q^{71}-7q^{69}+14q^{67}+12q^{65}-20q^{63}-25q^{61}+27q^{59}+41q^{57}-31q^{55}-61q^{53}+28q^{51}+77q^{49}-13q^{47}-88q^{45}-2q^{43}+86q^{41}+18q^{39}-68q^{37}-36q^{35}+49q^{33}+41q^{31}-21q^{29}-49q^{27}-2q^{25}+46q^{23}+28q^{21}-47q^{19}-48q^{17}+42q^{15}+65q^{13}-31q^{11}-77q^{9}+18q^{7}+84q^{5}+3q^{3}-83q-17q^{-1}+68q^{-3}+35q^{-5}-51q^{-7}-39q^{-9}+31q^{-11}+37q^{-13}-12q^{-15}-28q^{-17}-q^{-19}+17q^{-21}+3q^{-23}-7q^{-25}-3q^{-27}+2q^{-29}+2q^{-31}-q^{-33}$
4	$q^{132}-2q^{130}-q^{128}+3q^{126}+3q^{122}-9q^{120}-2q^{118}+15q^{116}+2q^{114}+2q^{112}-37q^{110}-12q^{108}+52q^{106}+34q^{104}+6q^{102}-110q^{100}-67q^{98}+104q^{96}+136q^{94}+65q^{92}-218q^{90}-218q^{88}+96q^{86}+285q^{84}+234q^{82}-252q^{80}-413q^{78}-53q^{76}+339q^{74}+441q^{72}-118q^{70}-474q^{68}-249q^{66}+208q^{64}+488q^{62}+91q^{60}-318q^{58}-331q^{56}-12q^{54}+338q^{52}+226q^{50}-75q^{48}-277q^{46}-179q^{44}+117q^{42}+270q^{40}+139q^{38}-185q^{36}-288q^{34}-77q^{32}+285q^{30}+306q^{28}-87q^{26}-357q^{24}-255q^{22}+259q^{20}+433q^{18}+52q^{16}-347q^{14}-415q^{12}+125q^{10}+454q^{8}+228q^{6}-200q^{4}-477q^{2}-81+315q^{-2}+323q^{-4}+27q^{-6}-353q^{-8}-210q^{-10}+81q^{-12}+246q^{-14}+165q^{-16}-137q^{-18}-169q^{-20}-65q^{-22}+83q^{-24}+134q^{-26}+q^{-28}-53q^{-30}-63q^{-32}-7q^{-34}+46q^{-36}+16q^{-38}+q^{-40}-17q^{-42}-10q^{-44}+7q^{-46}+3q^{-48}+3q^{-50}-2q^{-52}-2q^{-54}+q^{-56}$
5	$q^{195}-2q^{193}-q^{191}+3q^{189}-4q^{181}-q^{179}+11q^{177}+5q^{175}-11q^{173}-19q^{171}-12q^{169}+18q^{167}+47q^{165}+39q^{163}-34q^{161}-107q^{159}-80q^{157}+47q^{155}+184q^{153}+187q^{151}-26q^{149}-320q^{147}-368q^{145}-32q^{143}+450q^{141}+644q^{139}+219q^{137}-573q^{135}-1018q^{133}-544q^{131}+601q^{129}+1424q^{127}+1036q^{125}-455q^{123}-1776q^{121}-1638q^{119}+85q^{117}+1977q^{115}+2238q^{113}+458q^{111}-1899q^{109}-2695q^{107}-1115q^{105}+1552q^{103}+2912q^{101}+1698q^{99}-994q^{97}-2790q^{95}-2114q^{93}+328q^{91}+2378q^{89}+2303q^{87}+276q^{85}-1794q^{83}-2190q^{81}-786q^{79}+1118q^{77}+1950q^{75}+1119q^{73}-514q^{71}-1563q^{69}-1322q^{67}-45q^{65}+1228q^{63}+1443q^{61}+470q^{59}-897q^{57}-1548q^{55}-862q^{53}+655q^{51}+1664q^{49}+1205q^{47}-426q^{45}-1801q^{43}-1574q^{41}+191q^{39}+1909q^{37}+1948q^{35}+121q^{33}-1950q^{31}-2308q^{29}-525q^{27}+1848q^{25}+2601q^{23}+1002q^{21}-1535q^{19}-2749q^{17}-1518q^{15}+1055q^{13}+2680q^{11}+1945q^{9}-409q^{7}-2337q^{5}-2241q^{3}-257q+1801q^{-1}+2235q^{-3}+841q^{-5}-1092q^{-7}-1992q^{-9}-1223q^{-11}+417q^{-13}+1515q^{-15}+1320q^{-17}+150q^{-19}-954q^{-21}-1167q^{-23}-488q^{-25}+436q^{-27}+862q^{-29}+581q^{-31}-62q^{-33}-507q^{-35}-498q^{-37}-146q^{-39}+229q^{-41}+337q^{-43}+181q^{-45}-51q^{-47}-172q^{-49}-146q^{-51}-28q^{-53}+74q^{-55}+83q^{-57}+33q^{-59}-18q^{-61}-34q^{-63}-25q^{-65}-q^{-67}+17q^{-69}+10q^{-71}-3q^{-75}-3q^{-77}-3q^{-79}+2q^{-81}+2q^{-83}-q^{-85}$

A2 Invariants.

Weight	Invariant
1,0	$q^{22}-q^{18}+q^{16}-3q^{14}-q^{12}+4q^{6}+3q^{2}-q^{-2}+q^{-4}-q^{-6}$
1,1	$q^{60}-4q^{58}+10q^{56}-20q^{54}+38q^{52}-66q^{50}+100q^{48}-146q^{46}+198q^{44}-246q^{42}+284q^{40}-300q^{38}+289q^{36}-230q^{34}+132q^{32}-4q^{30}-149q^{28}+306q^{26}-448q^{24}+552q^{22}-611q^{20}+608q^{18}-560q^{16}+454q^{14}-320q^{12}+166q^{10}-2q^{8}-126q^{6}+239q^{4}-296q^{2}+326-308q^{-2}+262q^{-4}-208q^{-6}+148q^{-8}-96q^{-10}+54q^{-12}-28q^{-14}+12q^{-16}-4q^{-18}+q^{-20}$
2,0	$q^{56}-2q^{52}-q^{50}+2q^{48}+q^{46}-4q^{44}+7q^{40}-q^{38}-7q^{36}+2q^{34}+8q^{32}-6q^{28}+5q^{26}+q^{24}-9q^{22}-3q^{20}+q^{18}-5q^{16}-3q^{14}+8q^{12}+3q^{10}-2q^{8}+5q^{6}+11q^{4}-2q^{2}-6+5q^{-2}+3q^{-4}-5q^{-6}-3q^{-8}+2q^{-10}+2q^{-12}-2q^{-14}-q^{-16}+q^{-18}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{48}-2q^{46}+4q^{42}-7q^{40}-q^{38}+11q^{36}-8q^{34}-q^{32}+17q^{30}-8q^{28}-5q^{26}+7q^{24}-9q^{22}-10q^{20}-3q^{18}+3q^{16}-2q^{12}+14q^{10}+9q^{8}-10q^{6}+10q^{4}+6q^{2}-13+5q^{-2}+3q^{-4}-8q^{-6}+3q^{-8}+q^{-10}-2q^{-12}+q^{-14}$
1,0,0	$q^{29}+q^{25}-q^{23}+q^{21}-4q^{19}-q^{17}-3q^{15}+q^{11}+3q^{9}+4q^{7}+q^{5}+3q^{3}-q+q^{-1}-2q^{-3}+q^{-5}-q^{-7}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{62}-q^{58}+q^{54}-3q^{52}-7q^{50}+7q^{46}-q^{42}+16q^{40}+12q^{38}-4q^{36}-3q^{34}+3q^{32}-13q^{30}-22q^{28}-8q^{26}-4q^{24}-13q^{22}+19q^{18}+4q^{16}+4q^{14}+20q^{12}+13q^{10}-6q^{8}+q^{6}+8q^{4}-3q^{2}-10+3q^{-4}-5q^{-6}-2q^{-8}+3q^{-10}-q^{-14}+q^{-16}$
1,0,0,0	$q^{36}+q^{32}+q^{30}-q^{28}+q^{26}-4q^{24}-2q^{22}-3q^{20}-3q^{18}+q^{14}+4q^{12}+3q^{10}+5q^{8}+q^{6}+3q^{4}-q^{2}-2q^{-4}+q^{-6}-q^{-8}$

B2 Invariants.

Weight	Invariant
0,1	$q^{48}-2q^{46}+4q^{44}-6q^{42}+9q^{40}-11q^{38}+13q^{36}-14q^{34}+11q^{32}-9q^{30}+2q^{28}+3q^{26}-11q^{24}+17q^{22}-22q^{20}+25q^{18}-25q^{16}+24q^{14}-18q^{12}+14q^{10}-5q^{8}+8q^{4}-10q^{2}+13-13q^{-2}+13q^{-4}-10q^{-6}+7q^{-8}-5q^{-10}+2q^{-12}-q^{-14}$
1,0	$q^{78}-2q^{74}-2q^{72}+2q^{70}+5q^{68}-8q^{64}-6q^{62}+6q^{60}+12q^{58}+q^{56}-13q^{54}-6q^{52}+11q^{50}+14q^{48}-4q^{46}-13q^{44}-q^{42}+11q^{40}+2q^{38}-12q^{36}-8q^{34}+5q^{32}+5q^{30}-7q^{28}-9q^{26}+3q^{24}+10q^{22}-7q^{18}+q^{16}+14q^{14}+7q^{12}-9q^{10}-10q^{8}+9q^{6}+15q^{4}-14-7q^{-2}+9q^{-4}+10q^{-6}-4q^{-8}-9q^{-10}-2q^{-12}+5q^{-14}+3q^{-16}-2q^{-18}-2q^{-20}+q^{-24}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{66}-2q^{64}+2q^{62}-3q^{60}+5q^{58}-8q^{56}+6q^{54}-8q^{52}+12q^{50}-10q^{48}+10q^{46}-6q^{44}+12q^{42}-q^{40}+q^{36}-9q^{34}+8q^{32}-21q^{30}+8q^{28}-25q^{26}+17q^{24}-19q^{22}+19q^{20}-13q^{18}+20q^{16}-q^{14}+13q^{12}+q^{10}+q^{8}+8q^{6}-7q^{4}+7q^{2}-12+10q^{-2}-11q^{-4}+8q^{-6}-9q^{-8}+6q^{-10}-4q^{-12}+3q^{-14}-2q^{-16}+q^{-18}$

G2 Invariants.

Weight

Invariant

1,0

$q^{114}-2q^{112}+4q^{110}-6q^{108}+4q^{106}-3q^{104}-4q^{102}+13q^{100}-20q^{98}+27q^{96}-25q^{94}+15q^{92}+6q^{90}-29q^{88}+54q^{86}-62q^{84}+53q^{82}-29q^{80}-10q^{78}+49q^{76}-72q^{74}+73q^{72}-44q^{70}+3q^{68}+32q^{66}-53q^{64}+37q^{62}-7q^{60}-33q^{58}+56q^{56}-58q^{54}+23q^{52}+30q^{50}-81q^{48}+105q^{46}-99q^{44}+53q^{42}+8q^{40}-67q^{38}+103q^{36}-103q^{34}+79q^{32}-24q^{30}-28q^{28}+63q^{26}-64q^{24}+43q^{22}-34q^{18}+52q^{16}-35q^{14}+5q^{12}+41q^{10}-70q^{8}+77q^{6}-52q^{4}+5q^{2}+39-70q^{-2}+76q^{-4}-56q^{-6}+24q^{-8}+8q^{-10}-33q^{-12}+39q^{-14}-32q^{-16}+19q^{-18}-5q^{-20}-5q^{-22}+7q^{-24}-8q^{-26}+5q^{-28}-2q^{-30}+q^{-32}$

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 28"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$t^{3}-5t^{2}+12t-15+12t^{-1}-5t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{6}+z^{4}+z^{2}+1$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 51, -2 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$-q^{2}+3q-5+8q^{-1}-8q^{-2}+9q^{-3}-8q^{-4}+5q^{-5}-3q^{-6}+q^{-7}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$z^{2}a^{6}+a^{6}-2z^{4}a^{4}-5z^{2}a^{4}-4a^{4}+z^{6}a^{2}+4z^{4}a^{2}+7z^{2}a^{2}+5a^{2}-z^{4}-2z^{2}-1$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$z^{4}a^{8}-z^{2}a^{8}+3z^{5}a^{7}-4z^{3}a^{7}+2za^{7}+4z^{6}a^{6}-4z^{4}a^{6}+2z^{2}a^{6}-a^{6}+3z^{7}a^{5}+2z^{5}a^{5}-9z^{3}a^{5}+6za^{5}+z^{8}a^{4}+8z^{6}a^{4}-17z^{4}a^{4}+12z^{2}a^{4}-4a^{4}+6z^{7}a^{3}-5z^{5}a^{3}-7z^{3}a^{3}+6za^{3}+z^{8}a^{2}+7z^{6}a^{2}-19z^{4}a^{2}+14z^{2}a^{2}-5a^{2}+3z^{7}a-3z^{5}a-4z^{3}a+3za+3z^{6}-7z^{4}+5z^{2}-1+z^{5}a^{-1}-2z^{3}a^{-1}+za^{-1}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_29, 10_163, K11n87, ...}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {...}

Vassiliev invariants

V2 and V3:

(1, 0)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $4$ $0$ $8$ $-{\frac {34}{3}}$ $-{\frac {14}{3}}$ $0$ $32$ $0$ $0$ ${\frac {32}{3}}$ $0$ $-{\frac {136}{3}}$ $-{\frac {56}{3}}$ $-{\frac {2609}{30}}$ $-{\frac {622}{15}}$ ${\frac {1742}{45}}$ $-{\frac {79}{18}}$ ${\frac {271}{30}}$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$-2 is the signature of 9 28. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 5                   1 -1 3                 2   2 1               3 1   -2 -1             5 2     3 -3           4 4       0 -5         5 4         1 -7       3 4           1 -9     2 5             -3 -11   1 3               2 -13   2                 -2 -15 1                   1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=0$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{5}$ $r=1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)

$n$	$J_{n}$
2	$q^{7}-3q^{6}+10q^{4}-13q^{3}-7q^{2}+33q-23-26q^{-1}+61q^{-2}-26q^{-3}-49q^{-4}+78q^{-5}-20q^{-6}-63q^{-7}+77q^{-8}-10q^{-9}-59q^{-10}+56q^{-11}-38q^{-13}+27q^{-14}+3q^{-15}-15q^{-16}+8q^{-17}+q^{-18}-3q^{-19}+q^{-20}$
3	$-q^{15}+3q^{14}-5q^{12}-5q^{11}+13q^{10}+14q^{9}-23q^{8}-32q^{7}+29q^{6}+63q^{5}-29q^{4}-102q^{3}+17q^{2}+149q+4-187q^{-1}-49q^{-2}+235q^{-3}+85q^{-4}-253q^{-5}-144q^{-6}+281q^{-7}+181q^{-8}-276q^{-9}-234q^{-10}+282q^{-11}+256q^{-12}-258q^{-13}-282q^{-14}+235q^{-15}+284q^{-16}-196q^{-17}-274q^{-18}+150q^{-19}+252q^{-20}-110q^{-21}-206q^{-22}+62q^{-23}+166q^{-24}-35q^{-25}-116q^{-26}+13q^{-27}+77q^{-28}-5q^{-29}-44q^{-30}-q^{-31}+25q^{-32}-12q^{-34}+q^{-35}+4q^{-36}+q^{-37}-3q^{-38}+q^{-39}$
4	$q^{26}-3q^{25}+5q^{23}+5q^{21}-20q^{20}-7q^{19}+23q^{18}+15q^{17}+35q^{16}-73q^{15}-63q^{14}+33q^{13}+69q^{12}+168q^{11}-124q^{10}-211q^{9}-71q^{8}+101q^{7}+470q^{6}-43q^{5}-376q^{4}-362q^{3}-42q^{2}+850q+253-384q^{-1}-758q^{-2}-438q^{-3}+1127q^{-4}+681q^{-5}-158q^{-6}-1087q^{-7}-978q^{-8}+1195q^{-9}+1080q^{-10}+223q^{-11}-1261q^{-12}-1492q^{-13}+1093q^{-14}+1350q^{-15}+616q^{-16}-1282q^{-17}-1854q^{-18}+882q^{-19}+1453q^{-20}+940q^{-21}-1151q^{-22}-2007q^{-23}+586q^{-24}+1355q^{-25}+1142q^{-26}-850q^{-27}-1895q^{-28}+236q^{-29}+1036q^{-30}+1155q^{-31}-441q^{-32}-1498q^{-33}-44q^{-34}+579q^{-35}+930q^{-36}-85q^{-37}-939q^{-38}-146q^{-39}+187q^{-40}+570q^{-41}+76q^{-42}-453q^{-43}-95q^{-44}-2q^{-45}+256q^{-46}+76q^{-47}-170q^{-48}-24q^{-49}-34q^{-50}+85q^{-51}+33q^{-52}-54q^{-53}+4q^{-54}-16q^{-55}+21q^{-56}+8q^{-57}-15q^{-58}+4q^{-59}-3q^{-60}+4q^{-61}+q^{-62}-3q^{-63}+q^{-64}$
5	$-q^{40}+3q^{39}-5q^{37}+2q^{34}+13q^{33}+7q^{32}-23q^{31}-24q^{30}-9q^{29}+18q^{28}+64q^{27}+57q^{26}-32q^{25}-126q^{24}-127q^{23}-8q^{22}+185q^{21}+289q^{20}+124q^{19}-234q^{18}-502q^{17}-360q^{16}+176q^{15}+734q^{14}+767q^{13}+47q^{12}-928q^{11}-1284q^{10}-503q^{9}+947q^{8}+1871q^{7}+1217q^{6}-733q^{5}-2382q^{4}-2143q^{3}+178q^{2}+2771q+3150+661q^{-1}-2816q^{-2}-4201q^{-3}-1806q^{-4}+2675q^{-5}+5078q^{-6}+3015q^{-7}-2081q^{-8}-5826q^{-9}-4379q^{-10}+1444q^{-11}+6292q^{-12}+5552q^{-13}-482q^{-14}-6579q^{-15}-6752q^{-16}-339q^{-17}+6650q^{-18}+7623q^{-19}+1345q^{-20}-6618q^{-21}-8470q^{-22}-2104q^{-23}+6423q^{-24}+8998q^{-25}+2976q^{-26}-6159q^{-27}-9479q^{-28}-3621q^{-29}+5737q^{-30}+9649q^{-31}+4343q^{-32}-5186q^{-33}-9694q^{-34}-4894q^{-35}+4460q^{-36}+9408q^{-37}+5392q^{-38}-3553q^{-39}-8863q^{-40}-5726q^{-41}+2556q^{-42}+8004q^{-43}+5788q^{-44}-1483q^{-45}-6836q^{-46}-5651q^{-47}+506q^{-48}+5562q^{-49}+5112q^{-50}+313q^{-51}-4144q^{-52}-4437q^{-53}-854q^{-54}+2895q^{-55}+3532q^{-56}+1109q^{-57}-1787q^{-58}-2657q^{-59}-1115q^{-60}+1003q^{-61}+1809q^{-62}+971q^{-63}-466q^{-64}-1166q^{-65}-727q^{-66}+180q^{-67}+664q^{-68}+497q^{-69}-21q^{-70}-374q^{-71}-302q^{-72}-14q^{-73}+182q^{-74}+161q^{-75}+27q^{-76}-80q^{-77}-89q^{-78}-17q^{-79}+45q^{-80}+34q^{-81}-7q^{-83}-16q^{-84}-9q^{-85}+16q^{-86}+4q^{-87}-7q^{-88}+q^{-89}-3q^{-91}+4q^{-92}+q^{-93}-3q^{-94}+q^{-95}$
6	$q^{57}-3q^{56}+5q^{54}-7q^{51}+5q^{50}-13q^{49}-7q^{48}+32q^{47}+15q^{46}+9q^{45}-35q^{44}-9q^{43}-69q^{42}-47q^{41}+102q^{40}+117q^{39}+120q^{38}-45q^{37}-50q^{36}-328q^{35}-330q^{34}+84q^{33}+354q^{32}+613q^{31}+332q^{30}+199q^{29}-814q^{28}-1298q^{27}-729q^{26}+163q^{25}+1417q^{24}+1713q^{23}+1867q^{22}-541q^{21}-2659q^{20}-3156q^{19}-2042q^{18}+945q^{17}+3462q^{16}+5874q^{15}+2618q^{14}-2128q^{13}-6111q^{12}-7151q^{11}-3450q^{10}+2593q^{9}+10526q^{8}+9438q^{7}+3327q^{6}-5935q^{5}-12847q^{4}-12319q^{3}-4092q^{2}+11604q+17056+13880q^{-1}+573q^{-2}-14619q^{-3}-22321q^{-4}-16287q^{-5}+6070q^{-6}+20900q^{-7}+25831q^{-8}+12673q^{-9}-9865q^{-10}-28956q^{-11}-30012q^{-12}-4947q^{-13}+18792q^{-14}+34868q^{-15}+26341q^{-16}-113q^{-17}-30369q^{-18}-41162q^{-19}-17520q^{-20}+12345q^{-21}+39359q^{-22}+37834q^{-23}+10925q^{-24}-27986q^{-25}-48207q^{-26}-28343q^{-27}+4690q^{-28}+40377q^{-29}+45815q^{-30}+20514q^{-31}-24123q^{-32}-51840q^{-33}-36388q^{-34}-2294q^{-35}+39425q^{-36}+50779q^{-37}+28108q^{-38}-19698q^{-39}-52910q^{-40}-42170q^{-41}-8743q^{-42}+36600q^{-43}+53162q^{-44}+34408q^{-45}-13892q^{-46}-50924q^{-47}-45889q^{-48}-15587q^{-49}+30618q^{-50}+52020q^{-51}+39362q^{-52}-5668q^{-53}-44279q^{-54}-46153q^{-55}-22440q^{-56}+20573q^{-57}+45495q^{-58}+41019q^{-59}+4005q^{-60}-32331q^{-61}-40782q^{-62}-26670q^{-63}+8214q^{-64}+33253q^{-65}+36870q^{-66}+11602q^{-67}-17701q^{-68}-29660q^{-69}-25326q^{-70}-2067q^{-71}+18586q^{-72}+27045q^{-73}+13711q^{-74}-5476q^{-75}-16493q^{-76}-18597q^{-77}-6635q^{-78}+6751q^{-79}+15478q^{-80}+10541q^{-81}+841q^{-82}-6267q^{-83}-10291q^{-84}-5932q^{-85}+652q^{-86}+6698q^{-87}+5643q^{-88}+2030q^{-89}-1141q^{-90}-4205q^{-91}-3291q^{-92}-904q^{-93}+2196q^{-94}+2111q^{-95}+1183q^{-96}+295q^{-97}-1264q^{-98}-1271q^{-99}-667q^{-100}+603q^{-101}+537q^{-102}+379q^{-103}+321q^{-104}-288q^{-105}-361q^{-106}-268q^{-107}+176q^{-108}+84q^{-109}+55q^{-110}+138q^{-111}-53q^{-112}-78q^{-113}-78q^{-114}+65q^{-115}+2q^{-116}-11q^{-117}+40q^{-118}-11q^{-119}-10q^{-120}-19q^{-121}+23q^{-122}-q^{-123}-11q^{-124}+9q^{-125}-3q^{-126}-3q^{-128}+4q^{-129}+q^{-130}-3q^{-131}+q^{-132}$
7	$-q^{77}+3q^{76}-5q^{74}+7q^{71}-5q^{69}+13q^{68}-2q^{67}-23q^{66}-15q^{65}-9q^{64}+35q^{63}+37q^{62}+3q^{61}+48q^{60}-12q^{59}-93q^{58}-112q^{57}-123q^{56}+62q^{55}+181q^{54}+183q^{53}+291q^{52}+108q^{51}-218q^{50}-472q^{49}-749q^{48}-378q^{47}+200q^{46}+674q^{45}+1362q^{44}+1204q^{43}+412q^{42}-758q^{41}-2361q^{40}-2630q^{39}-1670q^{38}+104q^{37}+3047q^{36}+4635q^{35}+4352q^{34}+2038q^{33}-2974q^{32}-6867q^{31}-8261q^{30}-6202q^{29}+747q^{28}+8024q^{27}+13005q^{26}+13048q^{25}+4750q^{24}-6746q^{23}-17245q^{22}-21869q^{21}-14196q^{20}+967q^{19}+18712q^{18}+31318q^{17}+27719q^{16}+10541q^{15}-15230q^{14}-38842q^{13}-43597q^{12}-28150q^{11}+4392q^{10}+41509q^{9}+59441q^{8}+50828q^{7}+14529q^{6}-36503q^{5}-71811q^{4}-76194q^{3}-41156q^{2}+22305q+77588+100568q^{-1}+73445q^{-2}+1668q^{-3}-74387q^{-4}-121116q^{-5}-108325q^{-6}-33227q^{-7}+61444q^{-8}+134199q^{-9}+142082q^{-10}+70799q^{-11}-39263q^{-12}-139390q^{-13}-172112q^{-14}-109851q^{-15}+10404q^{-16}+135374q^{-17}+195649q^{-18}+148502q^{-19}+23194q^{-20}-124698q^{-21}-212770q^{-22}-183037q^{-23}-57485q^{-24}+108176q^{-25}+222542q^{-26}+213190q^{-27}+91190q^{-28}-89270q^{-29}-227473q^{-30}-237242q^{-31}-121219q^{-32}+68997q^{-33}+227690q^{-34}+256635q^{-35}+148013q^{-36}-50093q^{-37}-225958q^{-38}-271062q^{-39}-170188q^{-40}+32240q^{-41}+222248q^{-42}+282653q^{-43}+189379q^{-44}-16678q^{-45}-218279q^{-46}-291142q^{-47}-205389q^{-48}+1790q^{-49}+213003q^{-50}+298140q^{-51}+220128q^{-52}+12336q^{-53}-206785q^{-54}-302683q^{-55}-233293q^{-56}-27729q^{-57}+197579q^{-58}+304994q^{-59}+246035q^{-60}+44594q^{-61}-184961q^{-62}-303315q^{-63}-256959q^{-64}-63871q^{-65}+166889q^{-66}+296421q^{-67}+265617q^{-68}+84992q^{-69}-143322q^{-70}-282689q^{-71}-269630q^{-72}-106496q^{-73}+113998q^{-74}+260755q^{-75}+267199q^{-76}+126626q^{-77}-80406q^{-78}-231033q^{-79}-256636q^{-80}-142071q^{-81}+45353q^{-82}+193919q^{-83}+236775q^{-84}+151020q^{-85}-11527q^{-86}-152801q^{-87}-208950q^{-88}-150896q^{-89}-16897q^{-90}+110527q^{-91}+174334q^{-92}+142131q^{-93}+38068q^{-94}-71539q^{-95}-137113q^{-96}-125449q^{-97}-49806q^{-98}+38658q^{-99}+100353q^{-100}+103690q^{-101}+53101q^{-102}-14169q^{-103}-67912q^{-104}-79898q^{-105}-49284q^{-106}-1678q^{-107}+41674q^{-108}+57380q^{-109}+41122q^{-110}+9779q^{-111}-22693q^{-112}-38092q^{-113}-31149q^{-114}-12426q^{-115}+10356q^{-116}+23492q^{-117}+21687q^{-118}+11359q^{-119}-3443q^{-120}-13181q^{-121}-13772q^{-122}-8967q^{-123}+39q^{-124}+6883q^{-125}+8148q^{-126}+6185q^{-127}+1000q^{-128}-3226q^{-129}-4318q^{-130}-3912q^{-131}-1160q^{-132}+1349q^{-133}+2176q^{-134}+2319q^{-135}+838q^{-136}-574q^{-137}-963q^{-138}-1200q^{-139}-520q^{-140}+149q^{-141}+357q^{-142}+692q^{-143}+311q^{-144}-112q^{-145}-153q^{-146}-284q^{-147}-102q^{-148}+3q^{-149}-13q^{-150}+175q^{-151}+93q^{-152}-39q^{-153}-22q^{-154}-58q^{-155}+8q^{-156}+7q^{-157}-40q^{-158}+37q^{-159}+24q^{-160}-12q^{-161}-4q^{-162}-13q^{-163}+13q^{-164}+6q^{-165}-16q^{-166}+5q^{-167}+5q^{-168}-3q^{-169}-3q^{-171}+4q^{-172}+q^{-173}-3q^{-174}+q^{-175}$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[9, 28]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[11, 15, 12, 14], X[5, 13, 6, 12], X[13, 7, 14, 6], X[15, 18, 16, 1], X[9, 16, 10, 17], X[17, 10, 18, 11], X[7, 2, 8, 3]]
In[3]:=	GaussCode[Knot[9, 28]]
Out[3]=	GaussCode[-1, 9, -2, 1, -4, 5, -9, 2, -7, 8, -3, 4, -5, 3, -6, 7, -8, 6]
In[4]:=	DTCode[Knot[9, 28]]
Out[4]=	DTCode[4, 8, 12, 2, 16, 14, 6, 18, 10]
In[5]:=	br = BR[Knot[9, 28]]
Out[5]=	BR[4, {-1, -1, 2, -1, -3, 2, 2, -3, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 9}
In[7]:=	BraidIndex[Knot[9, 28]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[9, 28]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[9, 28]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 3, 3, {4, 6}, 1}
In[10]:=	alex = Alexander[Knot[9, 28]][t]
Out[10]=	-3 5 12 2 3 -15 + t - -- + -- + 12 t - 5 t + t 2 t t
In[11]:=	Conway[Knot[9, 28]][z]
Out[11]=	2 4 6 1 + z + z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 28], Knot[9, 29], Knot[10, 163], Knot[11, NonAlternating, 87]}
In[13]:=	{KnotDet[Knot[9, 28]], KnotSignature[Knot[9, 28]]}
Out[13]=	{51, -2}
In[14]:=	Jones[Knot[9, 28]][q]
Out[14]=	-7 3 5 8 9 8 8 2 -5 + q - -- + -- - -- + -- - -- + - + 3 q - q 6 5 4 3 2 q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 28]}
In[16]:=	A2Invariant[Knot[9, 28]][q]
Out[16]=	-22 -18 -16 3 -12 4 3 2 4 6 q - q + q - --- - q + -- + -- - q + q - q 14 6 2 q q q
In[17]:=	HOMFLYPT[Knot[9, 28]][a, z]
Out[17]=	2 4 6 2 2 2 4 2 6 2 4 -1 + 5 a - 4 a + a - 2 z + 7 a z - 5 a z + a z - z + 2 4 4 4 2 6 4 a z - 2 a z + a z
In[18]:=	Kauffman[Knot[9, 28]][a, z]
Out[18]=	2 4 6 z 3 5 7 2 -1 - 5 a - 4 a - a + - + 3 a z + 6 a z + 6 a z + 2 a z + 5 z + a 3 2 2 4 2 6 2 8 2 2 z 3 3 3 14 a z + 12 a z + 2 a z - a z - ---- - 4 a z - 7 a z - a 5 3 7 3 4 2 4 4 4 6 4 8 4 9 a z - 4 a z - 7 z - 19 a z - 17 a z - 4 a z + a z + 5 z 5 3 5 5 5 7 5 6 2 6 -- - 3 a z - 5 a z + 2 a z + 3 a z + 3 z + 7 a z + a 4 6 6 6 7 3 7 5 7 2 8 4 8 8 a z + 4 a z + 3 a z + 6 a z + 3 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[9, 28]], Vassiliev[3][Knot[9, 28]]}
Out[19]=	{1, 0}
In[20]:=	Kh[Knot[9, 28]][q, t]
Out[20]=	4 5 1 2 1 3 2 5 3 -- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 3 q 15 6 13 5 11 5 11 4 9 4 9 3 7 3 q q t q t q t q t q t q t q t 4 5 4 4 2 t 2 3 2 5 3 ----- + ----- + ---- + ---- + --- + 3 q t + q t + 2 q t + q t 7 2 5 2 5 3 q q t q t q t q t
In[21]:=	ColouredJones[Knot[9, 28], 2][q]
Out[21]=	-20 3 -18 8 15 3 27 38 56 59 -23 + q - --- + q + --- - --- + --- + --- - --- + --- - --- - 19 17 16 15 14 13 11 10 q q q q q q q q 10 77 63 20 78 49 26 61 26 2 3 -- + -- - -- - -- + -- - -- - -- + -- - -- + 33 q - 7 q - 13 q + 9 8 7 6 5 4 3 2 q q q q q q q q q 4 6 7 10 q - 3 q + q

See/edit the Rolfsen_Splice_Template. Back to the top.

9_27 9_29