10 121

10_120

10_122

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10 121 Further Notes and Views

Knot presentations

Planar diagram presentation	X1627 X7,20,8,1 X9,19,10,18 X3,11,4,10 X17,5,18,4 X5,12,6,13 X11,16,12,17 X19,14,20,15 X13,8,14,9 X15,2,16,3
Gauss code	-1, 10, -4, 5, -6, 1, -2, 9, -3, 4, -7, 6, -9, 8, -10, 7, -5, 3, -8, 2
Dowker-Thistlethwaite code	6 10 12 20 18 16 8 2 4 14
Conway Notation	[9*20]

Minimum Braid Representative: Length is 11, width is 4. Braid index is 4.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-10][-2] Hyperbolic Volume 16.9749 A-Polynomial See Data:10 121/A-polynomial

[edit Notes for 10 121'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 10 121's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-11t^{2}+27t-35+27t^{-1}-11t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}+z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 115, -2 }
Jones polynomial	$-q^{2}+5q-10+15q^{-1}-18q^{-2}+20q^{-3}-18q^{-4}+14q^{-5}-9q^{-6}+4q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-z^{4}a^{6}-z^{2}a^{6}-a^{6}+z^{6}a^{4}+2z^{4}a^{4}+3z^{2}a^{4}+2a^{4}+z^{6}a^{2}+z^{4}a^{2}-z^{2}a^{2}-a^{2}-z^{4}+1$
Kauffman polynomial (db, data sources)	$z^{5}a^{9}-z^{3}a^{9}+4z^{6}a^{8}-5z^{4}a^{8}+z^{2}a^{8}+8z^{7}a^{7}-13z^{5}a^{7}+8z^{3}a^{7}-2za^{7}+9z^{8}a^{6}-14z^{6}a^{6}+9z^{4}a^{6}-3z^{2}a^{6}+a^{6}+4z^{9}a^{5}+9z^{7}a^{5}-28z^{5}a^{5}+19z^{3}a^{5}-3za^{5}+19z^{8}a^{4}-36z^{6}a^{4}+22z^{4}a^{4}-7z^{2}a^{4}+2a^{4}+4z^{9}a^{3}+11z^{7}a^{3}-30z^{5}a^{3}+14z^{3}a^{3}-za^{3}+10z^{8}a^{2}-13z^{6}a^{2}+3z^{4}a^{2}-3z^{2}a^{2}+a^{2}+10z^{7}a-15z^{5}a+4z^{3}a+5z^{6}-5z^{4}+1+z^{5}a^{-1}$
The A2 invariant	$-q^{24}+2q^{22}-2q^{20}-2q^{18}+4q^{16}-3q^{14}+3q^{12}-q^{8}+3q^{6}-4q^{4}+4q^{2}-1-q^{-2}+3q^{-4}-q^{-6}$
The G2 invariant	$q^{128}-3q^{126}+7q^{124}-13q^{122}+16q^{120}-16q^{118}+7q^{116}+17q^{114}-48q^{112}+88q^{110}-120q^{108}+119q^{106}-76q^{104}-33q^{102}+190q^{100}-339q^{98}+424q^{96}-367q^{94}+144q^{92}+189q^{90}-524q^{88}+710q^{86}-650q^{84}+336q^{82}+111q^{80}-519q^{78}+707q^{76}-574q^{74}+195q^{72}+258q^{70}-566q^{68}+567q^{66}-274q^{64}-195q^{62}+623q^{60}-813q^{58}+696q^{56}-280q^{54}-274q^{52}+766q^{50}-1016q^{48}+928q^{46}-540q^{44}-20q^{42}+548q^{40}-851q^{38}+845q^{36}-520q^{34}+30q^{32}+417q^{30}-637q^{28}+517q^{26}-141q^{24}-311q^{22}+622q^{20}-634q^{18}+356q^{16}+94q^{14}-517q^{12}+736q^{10}-680q^{8}+389q^{6}-3q^{4}-331q^{2}+497-471q^{-2}+323q^{-4}-115q^{-6}-58q^{-8}+155q^{-10}-181q^{-12}+143q^{-14}-81q^{-16}+29q^{-18}+10q^{-20}-25q^{-22}+26q^{-24}-20q^{-26}+10q^{-28}-4q^{-30}+q^{-32}$

Further Quantum Invariants

Further quantum knot invariants for 10_121.

A1 Invariants.

Weight	Invariant
1	$-q^{17}+3q^{15}-5q^{13}+5q^{11}-4q^{9}+2q^{7}+2q^{5}-3q^{3}+5q-5q^{-1}+4q^{-3}-q^{-5}$
2	$q^{48}-3q^{46}+q^{44}+10q^{42}-18q^{40}-5q^{38}+44q^{36}-30q^{34}-40q^{32}+71q^{30}-9q^{28}-68q^{26}+53q^{24}+24q^{22}-53q^{20}+4q^{18}+40q^{16}-9q^{14}-45q^{12}+35q^{10}+38q^{8}-71q^{6}+8q^{4}+68q^{2}-53-21q^{-2}+53q^{-4}-14q^{-6}-21q^{-8}+15q^{-10}+q^{-12}-4q^{-14}+q^{-16}$
3	$-q^{93}+3q^{91}-q^{89}-6q^{87}+3q^{85}+17q^{83}-4q^{81}-53q^{79}-q^{77}+115q^{75}+51q^{73}-191q^{71}-179q^{69}+243q^{67}+366q^{65}-204q^{63}-573q^{61}+55q^{59}+731q^{57}+168q^{55}-768q^{53}-409q^{51}+679q^{49}+597q^{47}-501q^{45}-688q^{43}+279q^{41}+681q^{39}-47q^{37}-619q^{35}-151q^{33}+509q^{31}+336q^{29}-384q^{27}-507q^{25}+236q^{23}+655q^{21}-53q^{19}-761q^{17}-159q^{15}+782q^{13}+388q^{11}-694q^{9}-577q^{7}+507q^{5}+671q^{3}-262q-634q^{-1}+29q^{-3}+499q^{-5}+116q^{-7}-317q^{-9}-154q^{-11}+150q^{-13}+127q^{-15}-49q^{-17}-76q^{-19}+14q^{-21}+28q^{-23}+q^{-25}-11q^{-27}-q^{-29}+4q^{-31}-q^{-33}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{24}+2q^{22}-2q^{20}-2q^{18}+4q^{16}-3q^{14}+3q^{12}-q^{8}+3q^{6}-4q^{4}+4q^{2}-1-q^{-2}+3q^{-4}-q^{-6}$
1,1	$q^{68}-6q^{66}+20q^{64}-50q^{62}+111q^{60}-224q^{58}+412q^{56}-712q^{54}+1155q^{52}-1734q^{50}+2410q^{48}-3106q^{46}+3677q^{44}-3940q^{42}+3770q^{40}-3072q^{38}+1810q^{36}-94q^{34}-1888q^{32}+3880q^{30}-5664q^{28}+7012q^{26}-7730q^{24}+7754q^{22}-7064q^{20}+5760q^{18}-3990q^{16}+1976q^{14}+17q^{12}-1768q^{10}+3064q^{8}-3800q^{6}+3997q^{4}-3742q^{2}+3190-2472q^{-2}+1767q^{-4}-1174q^{-6}+712q^{-8}-388q^{-10}+194q^{-12}-88q^{-14}+32q^{-16}-8q^{-18}+q^{-20}$
2,0	$q^{62}-2q^{60}+5q^{56}-3q^{54}-9q^{52}+22q^{48}+2q^{46}-29q^{44}-2q^{42}+29q^{40}+q^{38}-36q^{36}+4q^{34}+29q^{32}-4q^{30}-23q^{28}+13q^{26}+13q^{24}-18q^{22}+8q^{20}+8q^{18}-14q^{16}-6q^{14}+25q^{12}-q^{10}-33q^{8}+9q^{6}+34q^{4}-10q^{2}-30+16q^{-2}+24q^{-4}-9q^{-6}-16q^{-8}+2q^{-10}+10q^{-12}-2q^{-14}-3q^{-16}+q^{-18}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{54}-3q^{52}+q^{50}+8q^{48}-14q^{46}+2q^{44}+26q^{42}-35q^{40}+q^{38}+45q^{36}-50q^{34}-q^{32}+46q^{30}-39q^{28}-7q^{26}+29q^{24}-6q^{22}-12q^{20}-q^{18}+24q^{16}-5q^{14}-35q^{12}+41q^{10}+7q^{8}-53q^{6}+43q^{4}+9q^{2}-41+29q^{-2}+5q^{-4}-19q^{-6}+11q^{-8}+2q^{-10}-4q^{-12}+q^{-14}$
1,0,0	$-q^{31}+2q^{29}-3q^{27}+q^{25}-3q^{23}+4q^{21}-3q^{19}+4q^{17}+q^{13}-q^{9}+2q^{7}-4q^{5}+4q^{3}-2q+3q^{-1}-2q^{-3}+3q^{-5}-q^{-7}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{68}-2q^{66}-2q^{64}+8q^{62}-16q^{58}+7q^{56}+23q^{54}-15q^{52}-25q^{50}+26q^{48}+23q^{46}-41q^{44}-20q^{42}+46q^{40}-q^{38}-48q^{36}+23q^{34}+39q^{32}-34q^{30}-15q^{28}+42q^{26}-7q^{24}-41q^{22}+27q^{20}+35q^{18}-41q^{16}-14q^{14}+51q^{12}-2q^{10}-45q^{8}+12q^{6}+31q^{4}-17q^{2}-19+18q^{-2}+12q^{-4}-12q^{-6}-3q^{-8}+9q^{-10}-q^{-12}-3q^{-14}+q^{-16}$
1,0,0,0	$-q^{38}+2q^{36}-3q^{34}-3q^{28}+4q^{26}-3q^{24}+4q^{22}+q^{20}+q^{18}+q^{16}-2q^{10}+2q^{8}-4q^{6}+4q^{4}-2q^{2}+2+2q^{-2}-2q^{-4}+3q^{-6}-q^{-8}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{54}+3q^{52}-7q^{50}+14q^{48}-24q^{46}+36q^{44}-50q^{42}+61q^{40}-65q^{38}+61q^{36}-50q^{34}+29q^{32}-2q^{30}-31q^{28}+65q^{26}-93q^{24}+116q^{22}-124q^{20}+123q^{18}-108q^{16}+83q^{14}-51q^{12}+17q^{10}+13q^{8}-39q^{6}+57q^{4}-65q^{2}+65-57q^{-2}+47q^{-4}-31q^{-6}+19q^{-8}-10q^{-10}+4q^{-12}-q^{-14}$
1,0	$q^{88}-3q^{84}-3q^{82}+4q^{80}+11q^{78}+q^{76}-19q^{74}-16q^{72}+18q^{70}+38q^{68}-q^{66}-53q^{64}-29q^{62}+47q^{60}+57q^{58}-21q^{56}-71q^{54}-11q^{52}+64q^{50}+35q^{48}-46q^{46}-47q^{44}+26q^{42}+48q^{40}-10q^{38}-47q^{36}+45q^{32}+11q^{30}-42q^{28}-21q^{26}+40q^{24}+35q^{22}-35q^{20}-50q^{18}+23q^{16}+65q^{14}+2q^{12}-68q^{10}-33q^{8}+54q^{6}+57q^{4}-24q^{2}-59-7q^{-2}+43q^{-4}+26q^{-6}-20q^{-8}-25q^{-10}+q^{-12}+15q^{-14}+6q^{-16}-4q^{-18}-4q^{-20}+q^{-24}$

D4 Invariants.

Weight

Invariant

1,0,0,0

$q^{74}-3q^{72}+4q^{70}-6q^{68}+12q^{66}-19q^{64}+23q^{62}-28q^{60}+41q^{58}-49q^{56}+48q^{54}-49q^{52}+52q^{50}-45q^{48}+27q^{46}-22q^{44}+7q^{42}+17q^{40}-40q^{38}+49q^{36}-65q^{34}+89q^{32}-91q^{30}+93q^{28}-95q^{26}+96q^{24}-76q^{22}+64q^{20}-55q^{18}+33q^{16}-7q^{14}-7q^{12}+16q^{10}-36q^{8}+49q^{6}-49q^{4}+51q^{2}-52+48q^{-2}-35q^{-4}+31q^{-6}-25q^{-8}+16q^{-10}-8q^{-12}+6q^{-14}-4q^{-16}+q^{-18}$

G2 Invariants.

Weight

Invariant

1,0

$q^{128}-3q^{126}+7q^{124}-13q^{122}+16q^{120}-16q^{118}+7q^{116}+17q^{114}-48q^{112}+88q^{110}-120q^{108}+119q^{106}-76q^{104}-33q^{102}+190q^{100}-339q^{98}+424q^{96}-367q^{94}+144q^{92}+189q^{90}-524q^{88}+710q^{86}-650q^{84}+336q^{82}+111q^{80}-519q^{78}+707q^{76}-574q^{74}+195q^{72}+258q^{70}-566q^{68}+567q^{66}-274q^{64}-195q^{62}+623q^{60}-813q^{58}+696q^{56}-280q^{54}-274q^{52}+766q^{50}-1016q^{48}+928q^{46}-540q^{44}-20q^{42}+548q^{40}-851q^{38}+845q^{36}-520q^{34}+30q^{32}+417q^{30}-637q^{28}+517q^{26}-141q^{24}-311q^{22}+622q^{20}-634q^{18}+356q^{16}+94q^{14}-517q^{12}+736q^{10}-680q^{8}+389q^{6}-3q^{4}-331q^{2}+497-471q^{-2}+323q^{-4}-115q^{-6}-58q^{-8}+155q^{-10}-181q^{-12}+143q^{-14}-81q^{-16}+29q^{-18}+10q^{-20}-25q^{-22}+26q^{-24}-20q^{-26}+10q^{-28}-4q^{-30}+q^{-32}$

.

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["10 121"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-11t^{2}+27t-35+27t^{-1}-11t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 115, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{2}+5q-10+15q^{-1}-18q^{-2}+20q^{-3}-18q^{-4}+14q^{-5}-9q^{-6}+4q^{-7}-q^{-8}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{4}a^{6}-z^{2}a^{6}-a^{6}+z^{6}a^{4}+2z^{4}a^{4}+3z^{2}a^{4}+2a^{4}+z^{6}a^{2}+z^{4}a^{2}-z^{2}a^{2}-a^{2}-z^{4}+1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{5}a^{9}-z^{3}a^{9}+4z^{6}a^{8}-5z^{4}a^{8}+z^{2}a^{8}+8z^{7}a^{7}-13z^{5}a^{7}+8z^{3}a^{7}-2za^{7}+9z^{8}a^{6}-14z^{6}a^{6}+9z^{4}a^{6}-3z^{2}a^{6}+a^{6}+4z^{9}a^{5}+9z^{7}a^{5}-28z^{5}a^{5}+19z^{3}a^{5}-3za^{5}+19z^{8}a^{4}-36z^{6}a^{4}+22z^{4}a^{4}-7z^{2}a^{4}+2a^{4}+4z^{9}a^{3}+11z^{7}a^{3}-30z^{5}a^{3}+14z^{3}a^{3}-za^{3}+10z^{8}a^{2}-13z^{6}a^{2}+3z^{4}a^{2}-3z^{2}a^{2}+a^{2}+10z^{7}a-15z^{5}a+4z^{3}a+5z^{6}-5z^{4}+1+z^{5}a^{-1}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a41, K11a183, K11a198, K11a331, ...}

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

Vassiliev invariants

V2 and V3:

(1, -2)

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 $4$ $-16$ $8$ ${\frac {110}{3}}$ $-{\frac {14}{3}}$ $-64$ $-{\frac {352}{3}}$ ${\frac {32}{3}}$ $-16$ ${\frac {32}{3}}$ $128$ ${\frac {440}{3}}$ $-{\frac {56}{3}}$ ${\frac {13471}{30}}$ $-{\frac {1102}{15}}$ ${\frac {9662}{45}}$ $-{\frac {511}{18}}$ ${\frac {511}{30}}$

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 10 121. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 5                     1 -1 3                   4   4 1                 6 1   -5 -1               9 4     5 -3             10 7       -3 -5           10 8         2 -7         8 10           2 -9       6 10             -4 -11     3 8               5 -13   1 6                 -5 -15   3                   3 -17 1                     -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-4$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=0$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{9}$ $r=1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{7}-5q^{6}+5q^{5}+15q^{4}-41q^{3}+12q^{2}+82q-115-20q^{-1}+203q^{-2}-175q^{-3}-99q^{-4}+312q^{-5}-178q^{-6}-179q^{-7}+348q^{-8}-129q^{-9}-215q^{-10}+291q^{-11}-52q^{-12}-186q^{-13}+170q^{-14}+7q^{-15}-106q^{-16}+59q^{-17}+17q^{-18}-32q^{-19}+10q^{-20}+4q^{-21}-4q^{-22}+q^{-23}$
3	$-q^{15}+5q^{14}-5q^{13}-10q^{12}+11q^{11}+32q^{10}-19q^{9}-100q^{8}+38q^{7}+208q^{6}+4q^{5}-404q^{4}-125q^{3}+641q^{2}+387q-874-788q^{-1}+1013q^{-2}+1320q^{-3}-1038q^{-4}-1872q^{-5}+896q^{-6}+2402q^{-7}-644q^{-8}-2813q^{-9}+294q^{-10}+3110q^{-11}+64q^{-12}-3232q^{-13}-449q^{-14}+3233q^{-15}+784q^{-16}-3059q^{-17}-1109q^{-18}+2765q^{-19}+1356q^{-20}-2331q^{-21}-1511q^{-22}+1798q^{-23}+1543q^{-24}-1233q^{-25}-1429q^{-26}+710q^{-27}+1184q^{-28}-297q^{-29}-866q^{-30}+34q^{-31}+556q^{-32}+72q^{-33}-296q^{-34}-89q^{-35}+134q^{-36}+60q^{-37}-54q^{-38}-25q^{-39}+18q^{-40}+8q^{-41}-5q^{-42}-4q^{-43}+4q^{-44}-q^{-45}$
4	$q^{26}-5q^{25}+5q^{24}+10q^{23}-16q^{22}-2q^{21}-25q^{20}+52q^{19}+82q^{18}-106q^{17}-97q^{16}-176q^{15}+289q^{14}+590q^{13}-173q^{12}-644q^{11}-1236q^{10}+481q^{9}+2406q^{8}+1127q^{7}-1182q^{6}-4720q^{5}-1616q^{4}+4894q^{3}+5955q^{2}+1523q-9653-8735q^{-1}+3822q^{-2}+12915q^{-3}+10561q^{-4}-11015q^{-5}-18889q^{-6}-4083q^{-7}+16524q^{-8}+23284q^{-9}-5597q^{-10}-26135q^{-11}-15946q^{-12}+13781q^{-13}+33559q^{-14}+3832q^{-15}-27397q^{-16}-26171q^{-17}+7018q^{-18}+38261q^{-19}+12749q^{-20}-24101q^{-21}-32180q^{-22}-530q^{-23}+37954q^{-24}+19437q^{-25}-18052q^{-26}-34142q^{-27}-7995q^{-28}+33219q^{-29}+23876q^{-30}-9405q^{-31}-31723q^{-32}-14982q^{-33}+23705q^{-34}+24646q^{-35}+813q^{-36}-23810q^{-37}-18806q^{-38}+11021q^{-39}+19624q^{-40}+8483q^{-41}-12097q^{-42}-16370q^{-43}+500q^{-44}+10327q^{-45}+9553q^{-46}-2283q^{-47}-9179q^{-48}-3362q^{-49}+2438q^{-50}+5538q^{-51}+1493q^{-52}-2867q^{-53}-2180q^{-54}-533q^{-55}+1685q^{-56}+1111q^{-57}-354q^{-58}-528q^{-59}-477q^{-60}+247q^{-61}+278q^{-62}-6q^{-63}-26q^{-64}-111q^{-65}+25q^{-66}+36q^{-67}-10q^{-68}+6q^{-69}-13q^{-70}+5q^{-71}+4q^{-72}-4q^{-73}+q^{-74}$
5	$-q^{40}+5q^{39}-5q^{38}-10q^{37}+16q^{36}+7q^{35}-5q^{34}-8q^{33}-34q^{32}-29q^{31}+90q^{30}+149q^{29}+4q^{28}-230q^{27}-408q^{26}-195q^{25}+497q^{24}+1219q^{23}+903q^{22}-872q^{21}-2763q^{20}-2813q^{19}+293q^{18}+5211q^{17}+7390q^{16}+2485q^{15}-7674q^{14}-14877q^{13}-10397q^{12}+7132q^{11}+25362q^{10}+25619q^{9}+298q^{8}-34927q^{7}-48831q^{6}-19965q^{5}+37974q^{4}+77330q^{3}+54320q^{2}-26923q-104297-102435q^{-1}-3736q^{-2}+120534q^{-3}+158188q^{-4}+55648q^{-5}-118030q^{-6}-211957q^{-7}-124190q^{-8}+91957q^{-9}+253523q^{-10}+201274q^{-11}-44019q^{-12}-275896q^{-13}-275665q^{-14}-19871q^{-15}+276333q^{-16}+338995q^{-17}+90509q^{-18}-257884q^{-19}-385285q^{-20}-159267q^{-21}+225845q^{-22}+414167q^{-23}+219600q^{-24}-187091q^{-25}-427285q^{-26}-269102q^{-27}+146528q^{-28}+429225q^{-29}+307561q^{-30}-107164q^{-31}-422549q^{-32}-337731q^{-33}+68646q^{-34}+410235q^{-35}+361491q^{-36}-29850q^{-37}-390960q^{-38}-380646q^{-39}-12099q^{-40}+363719q^{-41}+394335q^{-42}+57800q^{-43}-325057q^{-44}-399728q^{-45}-106875q^{-46}+273351q^{-47}+392638q^{-48}+155175q^{-49}-208778q^{-50}-368727q^{-51}-196351q^{-52}+135017q^{-53}+325835q^{-54}+223155q^{-55}-59569q^{-56}-265676q^{-57}-229518q^{-58}-7919q^{-59}+194245q^{-60}+213473q^{-61}+58443q^{-62}-121321q^{-63}-178297q^{-64}-86326q^{-65}+57493q^{-66}+131709q^{-67}+91411q^{-68}-10429q^{-69}-83915q^{-70}-79185q^{-71}-16583q^{-72}+43794q^{-73}+57614q^{-74}+26168q^{-75}-15987q^{-76}-35432q^{-77}-24141q^{-78}+1065q^{-79}+17892q^{-80}+16938q^{-81}+4466q^{-82}-6938q^{-83}-9673q^{-84}-4710q^{-85}+1710q^{-86}+4482q^{-87}+3074q^{-88}+140q^{-89}-1652q^{-90}-1582q^{-91}-417q^{-92}+523q^{-93}+628q^{-94}+230q^{-95}-100q^{-96}-196q^{-97}-131q^{-98}+32q^{-99}+78q^{-100}+10q^{-101}-7q^{-102}-q^{-103}-14q^{-104}-q^{-105}+13q^{-106}-5q^{-107}-4q^{-108}+4q^{-109}-q^{-110}$
6	Not Available
7	Not Available

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[10, 121]]
Out[2]=	PD[X[1, 6, 2, 7], X[7, 20, 8, 1], X[9, 19, 10, 18], X[3, 11, 4, 10], X[17, 5, 18, 4], X[5, 12, 6, 13], X[11, 16, 12, 17], X[19, 14, 20, 15], X[13, 8, 14, 9], X[15, 2, 16, 3]]
In[3]:=	GaussCode[Knot[10, 121]]
Out[3]=	GaussCode[-1, 10, -4, 5, -6, 1, -2, 9, -3, 4, -7, 6, -9, 8, -10, 7, -5, 3, -8, 2]
In[4]:=	DTCode[Knot[10, 121]]
Out[4]=	DTCode[6, 10, 12, 20, 18, 16, 8, 2, 4, 14]
In[5]:=	br = BR[Knot[10, 121]]
Out[5]=	BR[4, {-1, -1, -2, 3, -2, 1, -2, 3, -2, 3, -2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 121]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 121]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[10, 121]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 121]][t]
Out[10]=	2 11 27 2 3 -35 + -- - -- + -- + 27 t - 11 t + 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 121]][z]
Out[11]=	2 4 6 1 + z + z + 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 121], Knot[11, Alternating, 41], Knot[11, Alternating, 183], Knot[11, Alternating, 198], Knot[11, Alternating, 331]}
In[13]:=	{KnotDet[Knot[10, 121]], KnotSignature[Knot[10, 121]]}
Out[13]=	{115, -2}
In[14]:=	Jones[Knot[10, 121]][q]
Out[14]=	-8 4 9 14 18 20 18 15 2 -10 - q + -- - -- + -- - -- + -- - -- + -- + 5 q - q 7 6 5 4 3 2 q q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 121]}
In[16]:=	A2Invariant[Knot[10, 121]][q]
Out[16]=	-24 2 2 2 4 3 3 -8 3 4 4 -1 - q + --- - --- - --- + --- - --- + --- - q + -- - -- + -- - 22 20 18 16 14 12 6 4 2 q q q q q q q q q 2 4 6 q + 3 q - q
In[17]:=	HOMFLYPT[Knot[10, 121]][a, z]
Out[17]=	2 4 6 2 2 4 2 6 2 4 2 4 4 4 1 - a + 2 a - a - a z + 3 a z - a z - z + a z + 2 a z - 6 4 2 6 4 6 a z + a z + a z
In[18]:=	Kauffman[Knot[10, 121]][a, z]
Out[18]=	2 4 6 3 5 7 2 2 4 2 1 + a + 2 a + a - a z - 3 a z - 2 a z - 3 a z - 7 a z - 6 2 8 2 3 3 3 5 3 7 3 9 3 3 a z + a z + 4 a z + 14 a z + 19 a z + 8 a z - a z - 5 4 2 4 4 4 6 4 8 4 z 5 5 z + 3 a z + 22 a z + 9 a z - 5 a z + -- - 15 a z - a 3 5 5 5 7 5 9 5 6 2 6 4 6 30 a z - 28 a z - 13 a z + a z + 5 z - 13 a z - 36 a z - 6 6 8 6 7 3 7 5 7 7 7 14 a z + 4 a z + 10 a z + 11 a z + 9 a z + 8 a z + 2 8 4 8 6 8 3 9 5 9 10 a z + 19 a z + 9 a z + 4 a z + 4 a z
In[19]:=	{Vassiliev[2][Knot[10, 121]], Vassiliev[3][Knot[10, 121]]}
Out[19]=	{1, -2}
In[20]:=	Kh[Knot[10, 121]][q, t]
Out[20]=	7 9 1 3 1 6 3 8 6 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 q q t q t q t q t q t q t q t 10 8 10 10 8 10 4 t 2 ----- + ----- + ----- + ----- + ---- + ---- + --- + 6 q t + q t + 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t 3 2 5 3 4 q t + q t
In[21]:=	ColouredJones[Knot[10, 121], 2][q]
Out[21]=	-23 4 4 10 32 17 59 106 7 170 -115 + q - --- + --- + --- - --- + --- + --- - --- + --- + --- - 22 21 20 19 18 17 16 15 14 q q q q q q q q q 186 52 291 215 129 348 179 178 312 99 175 --- - --- + --- - --- - --- + --- - --- - --- + --- - -- - --- + 13 12 11 10 9 8 7 6 5 4 3 q q q q q q q q q q q 203 20 2 3 4 5 6 7 --- - -- + 82 q + 12 q - 41 q + 15 q + 5 q - 5 q + q 2 q q

See/edit the Rolfsen_Splice_Template. Back to the top.

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