10 102

10_101

10_103

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10 102 Further Notes and Views

Knot presentations

Planar diagram presentation	X6271 X16,10,17,9 X10,3,11,4 X2,15,3,16 X14,5,15,6 X18,8,19,7 X4,11,5,12 X8,18,9,17 X20,14,1,13 X12,20,13,19
Gauss code	1, -4, 3, -7, 5, -1, 6, -8, 2, -3, 7, -10, 9, -5, 4, -2, 8, -6, 10, -9
Dowker-Thistlethwaite code	6 10 14 18 16 4 20 2 8 12
Conway Notation	[3:2:20]

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

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Chiral Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-5][-7] Hyperbolic Volume 13.7273 A-Polynomial See Data:10 102/A-polynomial

[edit Notes for 10 102's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $3$ Rasmussen s-Invariant 0

[edit Notes for 10 102's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+8t^{2}-16t+21-16t^{-1}+8t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-4z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 73, 0 }
Jones polynomial	$q^{6}-3q^{5}+6q^{4}-9q^{3}+11q^{2}-12q+12-9q^{-1}+6q^{-2}-3q^{-3}+q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-2}-z^{6}+a^{2}z^{4}-3z^{4}a^{-2}+z^{4}a^{-4}-3z^{4}+2a^{2}z^{2}-3z^{2}a^{-2}+2z^{2}a^{-4}-3z^{2}+a^{2}-a^{-2}+a^{-4}$
Kauffman polynomial (db, data sources)	$2z^{9}a^{-1}+2z^{9}a^{-3}+9z^{8}a^{-2}+4z^{8}a^{-4}+5z^{8}+6az^{7}+4z^{7}a^{-1}+z^{7}a^{-3}+3z^{7}a^{-5}+5a^{2}z^{6}-24z^{6}a^{-2}-11z^{6}a^{-4}+z^{6}a^{-6}-7z^{6}+3a^{3}z^{5}-9az^{5}-17z^{5}a^{-1}-14z^{5}a^{-3}-9z^{5}a^{-5}+a^{4}z^{4}-6a^{2}z^{4}+21z^{4}a^{-2}+8z^{4}a^{-4}-3z^{4}a^{-6}+3z^{4}-3a^{3}z^{3}+7az^{3}+16z^{3}a^{-1}+13z^{3}a^{-3}+7z^{3}a^{-5}-a^{4}z^{2}+3a^{2}z^{2}-8z^{2}a^{-2}-4z^{2}a^{-4}+2z^{2}a^{-6}+2z^{2}-2az-4za^{-1}-4za^{-3}-2za^{-5}-a^{2}+a^{-2}+a^{-4}$
The A2 invariant	$q^{12}-q^{10}+q^{8}+q^{6}-2q^{4}+3q^{2}-1+q^{-2}-2q^{-6}+2q^{-8}-2q^{-10}+q^{-12}+q^{-14}-q^{-16}+q^{-18}$
The G2 invariant	$q^{66}-2q^{64}+4q^{62}-6q^{60}+5q^{58}-3q^{56}-2q^{54}+11q^{52}-18q^{50}+26q^{48}-28q^{46}+19q^{44}-4q^{42}-19q^{40}+42q^{38}-59q^{36}+68q^{34}-61q^{32}+33q^{30}+15q^{28}-64q^{26}+110q^{24}-123q^{22}+100q^{20}-42q^{18}-40q^{16}+108q^{14}-134q^{12}+108q^{10}-29q^{8}-57q^{6}+110q^{4}-105q^{2}+39+57q^{-2}-141q^{-4}+164q^{-6}-116q^{-8}+14q^{-10}+107q^{-12}-193q^{-14}+216q^{-16}-165q^{-18}+60q^{-20}+55q^{-22}-151q^{-24}+192q^{-26}-166q^{-28}+88q^{-30}+14q^{-32}-100q^{-34}+135q^{-36}-108q^{-38}+29q^{-40}+61q^{-42}-125q^{-44}+126q^{-46}-65q^{-48}-34q^{-50}+131q^{-52}-171q^{-54}+148q^{-56}-68q^{-58}-33q^{-60}+111q^{-62}-142q^{-64}+125q^{-66}-69q^{-68}+6q^{-70}+42q^{-72}-63q^{-74}+57q^{-76}-36q^{-78}+16q^{-80}+q^{-82}-10q^{-84}+10q^{-86}-9q^{-88}+5q^{-90}-2q^{-92}+q^{-94}$

Further Quantum Invariants

Further quantum knot invariants for 10_102.

A1 Invariants.

Weight	Invariant
1	$q^{9}-2q^{7}+3q^{5}-3q^{3}+3q-q^{-3}+2q^{-5}-3q^{-7}+3q^{-9}-2q^{-11}+q^{-13}$
2	$q^{26}-2q^{24}+5q^{20}-7q^{18}+q^{16}+13q^{14}-17q^{12}-3q^{10}+25q^{8}-17q^{6}-13q^{4}+24q^{2}-2-15q^{-2}+6q^{-4}+13q^{-6}-9q^{-8}-12q^{-10}+20q^{-12}+q^{-14}-24q^{-16}+16q^{-18}+13q^{-20}-24q^{-22}+4q^{-24}+17q^{-26}-12q^{-28}-5q^{-30}+8q^{-32}-q^{-34}-2q^{-36}+q^{-38}$
3	$q^{51}-2q^{49}+2q^{45}+q^{43}-3q^{41}-q^{39}+6q^{37}-5q^{35}-10q^{33}+14q^{31}+24q^{29}-23q^{27}-53q^{25}+28q^{23}+91q^{21}-13q^{19}-129q^{17}-24q^{15}+148q^{13}+70q^{11}-140q^{9}-107q^{7}+99q^{5}+134q^{3}-46q-131q^{-1}-6q^{-3}+111q^{-5}+57q^{-7}-87q^{-9}-90q^{-11}+58q^{-13}+118q^{-15}-35q^{-17}-136q^{-19}+5q^{-21}+148q^{-23}+29q^{-25}-150q^{-27}-67q^{-29}+135q^{-31}+105q^{-33}-98q^{-35}-133q^{-37}+47q^{-39}+140q^{-41}+4q^{-43}-116q^{-45}-47q^{-47}+77q^{-49}+65q^{-51}-36q^{-53}-56q^{-55}+3q^{-57}+36q^{-59}+10q^{-61}-17q^{-63}-8q^{-65}+5q^{-67}+4q^{-69}-q^{-71}-2q^{-73}+q^{-75}$
4	$q^{84}-2q^{82}+2q^{78}-2q^{76}+5q^{74}-5q^{72}-q^{70}+q^{68}-11q^{66}+21q^{64}+5q^{62}+5q^{60}-18q^{58}-67q^{56}+25q^{54}+70q^{52}+104q^{50}-17q^{48}-250q^{46}-139q^{44}+118q^{42}+426q^{40}+261q^{38}-418q^{36}-636q^{34}-223q^{32}+708q^{30}+945q^{28}-70q^{26}-1017q^{24}-1010q^{22}+343q^{20}+1386q^{18}+739q^{16}-626q^{14}-1428q^{12}-481q^{10}+947q^{8}+1150q^{6}+222q^{4}-987q^{2}-928+87q^{-2}+845q^{-4}+724q^{-6}-234q^{-8}-813q^{-10}-493q^{-12}+369q^{-14}+818q^{-16}+262q^{-18}-620q^{-20}-779q^{-22}+102q^{-24}+874q^{-26}+588q^{-28}-530q^{-30}-1064q^{-32}-171q^{-34}+923q^{-36}+1002q^{-38}-243q^{-40}-1256q^{-42}-685q^{-44}+591q^{-46}+1318q^{-48}+423q^{-50}-934q^{-52}-1115q^{-54}-211q^{-56}+1022q^{-58}+978q^{-60}-69q^{-62}-868q^{-64}-845q^{-66}+174q^{-68}+789q^{-70}+566q^{-72}-105q^{-74}-684q^{-76}-390q^{-78}+129q^{-80}+441q^{-82}+324q^{-84}-144q^{-86}-277q^{-88}-183q^{-90}+58q^{-92}+199q^{-94}+72q^{-96}-25q^{-98}-93q^{-100}-47q^{-102}+32q^{-104}+26q^{-106}+19q^{-108}-11q^{-110}-14q^{-112}+2q^{-114}+q^{-116}+4q^{-118}-q^{-120}-2q^{-122}+q^{-124}$

A2 Invariants.

Weight	Invariant
1,0	$q^{12}-q^{10}+q^{8}+q^{6}-2q^{4}+3q^{2}-1+q^{-2}-2q^{-6}+2q^{-8}-2q^{-10}+q^{-12}+q^{-14}-q^{-16}+q^{-18}$
1,1	$q^{36}-4q^{34}+10q^{32}-20q^{30}+36q^{28}-58q^{26}+90q^{24}-128q^{22}+171q^{20}-218q^{18}+278q^{16}-348q^{14}+407q^{12}-458q^{10}+496q^{8}-480q^{6}+399q^{4}-242q^{2}+24+242q^{-2}-534q^{-4}+798q^{-6}-1008q^{-8}+1140q^{-10}-1165q^{-12}+1098q^{-14}-934q^{-16}+698q^{-18}-408q^{-20}+96q^{-22}+188q^{-24}-420q^{-26}+583q^{-28}-656q^{-30}+638q^{-32}-556q^{-34}+445q^{-36}-324q^{-38}+208q^{-40}-122q^{-42}+66q^{-44}-30q^{-46}+12q^{-48}-4q^{-50}+q^{-52}$
2,0	$q^{32}-q^{30}-q^{28}+3q^{26}-4q^{22}+2q^{20}+7q^{18}-2q^{16}-10q^{14}+4q^{12}+12q^{10}-9q^{8}-9q^{6}+9q^{4}+4q^{2}-5-3q^{-2}+8q^{-4}-2q^{-6}-4q^{-8}+7q^{-10}+q^{-12}-7q^{-14}+4q^{-16}+9q^{-18}-8q^{-20}-5q^{-22}+6q^{-24}+6q^{-26}-7q^{-28}-7q^{-30}+8q^{-32}+3q^{-34}-5q^{-36}-3q^{-38}+2q^{-40}+3q^{-42}-q^{-44}-q^{-46}+q^{-48}$

A3 Invariants.

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

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{34}-q^{32}-q^{30}+3q^{28}-4q^{24}+q^{22}+7q^{20}-9q^{16}+3q^{14}+13q^{12}-7q^{10}-12q^{8}+14q^{6}+7q^{4}-12q^{2}+1+15q^{-2}-3q^{-4}-12q^{-6}+8q^{-8}+5q^{-10}-19q^{-12}-3q^{-14}+17q^{-16}-11q^{-18}-12q^{-20}+17q^{-22}+9q^{-24}-12q^{-26}-q^{-28}+14q^{-30}-10q^{-34}+3q^{-36}+6q^{-38}-6q^{-40}-2q^{-42}+4q^{-44}-q^{-46}-q^{-48}+q^{-50}$
1,0,0,0	$q^{18}-q^{16}+2q^{14}+2q^{8}-2q^{6}+3q^{4}-q^{2}+1-q^{-6}-q^{-8}-2q^{-12}+2q^{-14}-2q^{-16}+2q^{-18}+2q^{-24}-q^{-26}+q^{-28}$

B2 Invariants.

Weight	Invariant
0,1	$q^{28}-2q^{26}+4q^{24}-7q^{22}+10q^{20}-14q^{18}+19q^{16}-21q^{14}+24q^{12}-22q^{10}+18q^{8}-10q^{6}+12q^{2}-24+34q^{-2}-42q^{-4}+46q^{-6}-46q^{-8}+41q^{-10}-32q^{-12}+22q^{-14}-9q^{-16}-3q^{-18}+13q^{-20}-20q^{-22}+23q^{-24}-24q^{-26}+23q^{-28}-19q^{-30}+14q^{-32}-9q^{-34}+5q^{-36}-2q^{-38}+q^{-40}$
1,0	$q^{46}-2q^{42}-2q^{40}+2q^{38}+6q^{36}+q^{34}-8q^{32}-8q^{30}+5q^{28}+16q^{26}+4q^{24}-16q^{22}-16q^{20}+9q^{18}+24q^{16}+3q^{14}-23q^{12}-13q^{10}+17q^{8}+20q^{6}-9q^{4}-20q^{2}+3+20q^{-2}+3q^{-4}-17q^{-6}-7q^{-8}+13q^{-10}+9q^{-12}-12q^{-14}-12q^{-16}+10q^{-18}+15q^{-20}-7q^{-22}-21q^{-24}+24q^{-28}+12q^{-30}-20q^{-32}-22q^{-34}+11q^{-36}+26q^{-38}+3q^{-40}-21q^{-42}-12q^{-44}+13q^{-46}+14q^{-48}-4q^{-50}-11q^{-52}-2q^{-54}+6q^{-56}+3q^{-58}-2q^{-60}-2q^{-62}+q^{-66}$

D4 Invariants.

Weight

Invariant

1,0,0,0

$q^{38}-2q^{36}+2q^{34}-3q^{32}+6q^{30}-8q^{28}+8q^{26}-10q^{24}+16q^{22}-16q^{20}+15q^{18}-17q^{16}+21q^{14}-15q^{12}+11q^{10}-9q^{8}+5q^{6}+7q^{4}-10q^{2}+17-24q^{-2}+31q^{-4}-32q^{-6}+33q^{-8}-39q^{-10}+33q^{-12}-32q^{-14}+26q^{-16}-24q^{-18}+14q^{-20}-7q^{-22}+3q^{-24}+5q^{-26}-9q^{-28}+18q^{-30}-16q^{-32}+19q^{-34}-19q^{-36}+20q^{-38}-17q^{-40}+13q^{-42}-12q^{-44}+8q^{-46}-5q^{-48}+3q^{-50}-2q^{-52}+q^{-54}$

G2 Invariants.

Weight

Invariant

1,0

$q^{66}-2q^{64}+4q^{62}-6q^{60}+5q^{58}-3q^{56}-2q^{54}+11q^{52}-18q^{50}+26q^{48}-28q^{46}+19q^{44}-4q^{42}-19q^{40}+42q^{38}-59q^{36}+68q^{34}-61q^{32}+33q^{30}+15q^{28}-64q^{26}+110q^{24}-123q^{22}+100q^{20}-42q^{18}-40q^{16}+108q^{14}-134q^{12}+108q^{10}-29q^{8}-57q^{6}+110q^{4}-105q^{2}+39+57q^{-2}-141q^{-4}+164q^{-6}-116q^{-8}+14q^{-10}+107q^{-12}-193q^{-14}+216q^{-16}-165q^{-18}+60q^{-20}+55q^{-22}-151q^{-24}+192q^{-26}-166q^{-28}+88q^{-30}+14q^{-32}-100q^{-34}+135q^{-36}-108q^{-38}+29q^{-40}+61q^{-42}-125q^{-44}+126q^{-46}-65q^{-48}-34q^{-50}+131q^{-52}-171q^{-54}+148q^{-56}-68q^{-58}-33q^{-60}+111q^{-62}-142q^{-64}+125q^{-66}-69q^{-68}+6q^{-70}+42q^{-72}-63q^{-74}+57q^{-76}-36q^{-78}+16q^{-80}+q^{-82}-10q^{-84}+10q^{-86}-9q^{-88}+5q^{-90}-2q^{-92}+q^{-94}$

.

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["10 102"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{3}+8t^{2}-16t+21-16t^{-1}+8t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 73, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{6}-3q^{5}+6q^{4}-9q^{3}+11q^{2}-12q+12-9q^{-1}+6q^{-2}-3q^{-3}+q^{-4}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{9}a^{-1}+2z^{9}a^{-3}+9z^{8}a^{-2}+4z^{8}a^{-4}+5z^{8}+6az^{7}+4z^{7}a^{-1}+z^{7}a^{-3}+3z^{7}a^{-5}+5a^{2}z^{6}-24z^{6}a^{-2}-11z^{6}a^{-4}+z^{6}a^{-6}-7z^{6}+3a^{3}z^{5}-9az^{5}-17z^{5}a^{-1}-14z^{5}a^{-3}-9z^{5}a^{-5}+a^{4}z^{4}-6a^{2}z^{4}+21z^{4}a^{-2}+8z^{4}a^{-4}-3z^{4}a^{-6}+3z^{4}-3a^{3}z^{3}+7az^{3}+16z^{3}a^{-1}+13z^{3}a^{-3}+7z^{3}a^{-5}-a^{4}z^{2}+3a^{2}z^{2}-8z^{2}a^{-2}-4z^{2}a^{-4}+2z^{2}a^{-6}+2z^{2}-2az-4za^{-1}-4za^{-3}-2za^{-5}-a^{2}+a^{-2}+a^{-4}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {...}

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

Vassiliev invariants

V2 and V3:

(-2, -1)

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 $-8$ $-8$ $32$ ${\frac {212}{3}}$ ${\frac {148}{3}}$ $64$ ${\frac {496}{3}}$ ${\frac {64}{3}}$ $88$ $-{\frac {256}{3}}$ $32$ $-{\frac {1696}{3}}$ $-{\frac {1184}{3}}$ $-{\frac {5431}{15}}$ ${\frac {5044}{15}}$ $-{\frac {37924}{45}}$ ${\frac {1831}{9}}$ $-{\frac {3511}{15}}$

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=$0 is the signature of 10 102. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{18}-3q^{17}+q^{16}+10q^{15}-16q^{14}-6q^{13}+39q^{12}-29q^{11}-34q^{10}+76q^{9}-26q^{8}-74q^{7}+101q^{6}-7q^{5}-106q^{4}+104q^{3}+15q^{2}-113q+83+28q^{-1}-87q^{-2}+46q^{-3}+24q^{-4}-45q^{-5}+18q^{-6}+10q^{-7}-15q^{-8}+6q^{-9}+2q^{-10}-3q^{-11}+q^{-12}$
3	$q^{36}-3q^{35}+q^{34}+5q^{33}+2q^{32}-16q^{31}-8q^{30}+32q^{29}+28q^{28}-49q^{27}-67q^{26}+52q^{25}+129q^{24}-37q^{23}-191q^{22}-17q^{21}+249q^{20}+99q^{19}-284q^{18}-197q^{17}+284q^{16}+302q^{15}-254q^{14}-399q^{13}+201q^{12}+481q^{11}-135q^{10}-542q^{9}+60q^{8}+582q^{7}+18q^{6}-602q^{5}-88q^{4}+585q^{3}+162q^{2}-548q-205+460q^{-1}+247q^{-2}-368q^{-3}-240q^{-4}+254q^{-5}+214q^{-6}-158q^{-7}-162q^{-8}+82q^{-9}+109q^{-10}-42q^{-11}-58q^{-12}+19q^{-13}+28q^{-14}-12q^{-15}-11q^{-16}+9q^{-17}+4q^{-18}-7q^{-19}+2q^{-21}+2q^{-22}-3q^{-23}+q^{-24}$
4	$q^{60}-3q^{59}+q^{58}+5q^{57}-3q^{56}+2q^{55}-19q^{54}+4q^{53}+35q^{52}+4q^{51}+8q^{50}-98q^{49}-42q^{48}+103q^{47}+101q^{46}+135q^{45}-239q^{44}-283q^{43}+9q^{42}+234q^{41}+603q^{40}-122q^{39}-595q^{38}-510q^{37}-60q^{36}+1182q^{35}+549q^{34}-372q^{33}-1125q^{32}-1079q^{31}+1159q^{30}+1348q^{29}+675q^{28}-1081q^{27}-2312q^{26}+255q^{25}+1529q^{24}+2032q^{23}-186q^{22}-3039q^{21}-1021q^{20}+958q^{19}+3045q^{18}+1059q^{17}-3118q^{16}-2115q^{15}+65q^{14}+3579q^{13}+2177q^{12}-2832q^{11}-2887q^{10}-816q^{9}+3738q^{8}+3059q^{7}-2276q^{6}-3336q^{5}-1678q^{4}+3418q^{3}+3638q^{2}-1318q-3215-2436q^{-1}+2403q^{-2}+3579q^{-3}-109q^{-4}-2287q^{-5}-2639q^{-6}+975q^{-7}+2632q^{-8}+693q^{-9}-922q^{-10}-1992q^{-11}-68q^{-12}+1279q^{-13}+686q^{-14}+25q^{-15}-977q^{-16}-305q^{-17}+348q^{-18}+273q^{-19}+243q^{-20}-298q^{-21}-140q^{-22}+40q^{-23}+16q^{-24}+132q^{-25}-65q^{-26}-19q^{-27}+6q^{-28}-29q^{-29}+40q^{-30}-16q^{-31}+4q^{-32}+6q^{-33}-13q^{-34}+8q^{-35}-4q^{-36}+2q^{-37}+2q^{-38}-3q^{-39}+q^{-40}$
5	$q^{90}-3q^{89}+q^{88}+5q^{87}-3q^{86}-3q^{85}-q^{84}-7q^{83}+6q^{82}+30q^{81}+8q^{80}-30q^{79}-45q^{78}-50q^{77}+16q^{76}+128q^{75}+157q^{74}+17q^{73}-192q^{72}-335q^{71}-235q^{70}+179q^{69}+605q^{68}+637q^{67}+82q^{66}-758q^{65}-1232q^{64}-757q^{63}+550q^{62}+1791q^{61}+1853q^{60}+283q^{59}-1920q^{58}-3045q^{57}-1897q^{56}+1146q^{55}+3922q^{54}+3977q^{53}+714q^{52}-3730q^{51}-5956q^{50}-3646q^{49}+2126q^{48}+7099q^{47}+6977q^{46}+1015q^{45}-6689q^{44}-9976q^{43}-5332q^{42}+4510q^{41}+11851q^{40}+10028q^{39}-659q^{38}-12077q^{37}-14368q^{36}-4326q^{35}+10632q^{34}+17699q^{33}+9687q^{32}-7753q^{31}-19725q^{30}-14815q^{29}+4003q^{28}+20514q^{27}+19230q^{26}+44q^{25}-20330q^{24}-22779q^{23}-3923q^{22}+19550q^{21}+25525q^{20}+7395q^{19}-18533q^{18}-27651q^{17}-10427q^{16}+17472q^{15}+29344q^{14}+13141q^{13}-16275q^{12}-30781q^{11}-15790q^{10}+14891q^{9}+31811q^{8}+18420q^{7}-12795q^{6}-32257q^{5}-21220q^{4}+10011q^{3}+31695q^{2}+23621q-6124-29716q^{-1}-25525q^{-2}+1714q^{-3}+26177q^{-4}+25991q^{-5}+2937q^{-6}-21127q^{-7}-24932q^{-8}-6866q^{-9}+15224q^{-10}+22028q^{-11}+9535q^{-12}-9293q^{-13}-17801q^{-14}-10441q^{-15}+4184q^{-16}+12919q^{-17}+9782q^{-18}-549q^{-19}-8329q^{-20}-7902q^{-21}-1488q^{-22}+4561q^{-23}+5657q^{-24}+2170q^{-25}-2047q^{-26}-3534q^{-27}-1961q^{-28}+586q^{-29}+1934q^{-30}+1431q^{-31}+42q^{-32}-923q^{-33}-875q^{-34}-207q^{-35}+370q^{-36}+459q^{-37}+187q^{-38}-109q^{-39}-225q^{-40}-124q^{-41}+41q^{-42}+84q^{-43}+50q^{-44}+13q^{-45}-32q^{-46}-40q^{-47}+8q^{-48}+13q^{-49}-3q^{-50}+10q^{-51}+q^{-52}-11q^{-53}+2q^{-54}+4q^{-55}-4q^{-56}+2q^{-57}+2q^{-58}-3q^{-59}+q^{-60}$
6	$q^{126}-3q^{125}+q^{124}+5q^{123}-3q^{122}-3q^{121}-6q^{120}+11q^{119}-5q^{118}+q^{117}+33q^{116}-11q^{115}-30q^{114}-59q^{113}+12q^{112}+7q^{111}+50q^{110}+184q^{109}+62q^{108}-79q^{107}-318q^{106}-221q^{105}-224q^{104}+66q^{103}+714q^{102}+764q^{101}+506q^{100}-475q^{99}-950q^{98}-1696q^{97}-1349q^{96}+541q^{95}+2130q^{94}+3217q^{93}+2014q^{92}+358q^{91}-3476q^{90}-5774q^{89}-4408q^{88}-654q^{87}+4951q^{86}+7925q^{85}+9048q^{84}+2364q^{83}-6422q^{82}-12597q^{81}-13210q^{80}-5607q^{79}+5394q^{78}+19052q^{77}+20077q^{76}+10828q^{75}-6186q^{74}-22897q^{73}-28992q^{72}-20834q^{71}+6473q^{70}+29024q^{69}+40296q^{68}+28989q^{67}-1052q^{66}-36130q^{65}-56952q^{64}-39274q^{63}-1927q^{62}+45337q^{61}+70330q^{60}+56932q^{59}+3884q^{58}-60852q^{57}-87335q^{56}-70063q^{55}-2028q^{54}+73356q^{53}+113490q^{52}+80747q^{51}-9705q^{50}-93783q^{49}-133650q^{48}-84650q^{47}+21804q^{46}+127695q^{45}+151189q^{44}+73937q^{43}-48988q^{42}-156602q^{41}-160004q^{40}-58160q^{39}+95729q^{38}+184571q^{37}+150009q^{36}+19514q^{35}-139593q^{34}-202709q^{33}-130123q^{32}+44450q^{31}+184290q^{30}+197880q^{29}+80015q^{28}-107233q^{27}-217422q^{26}-177844q^{25}+117q^{24}+171609q^{23}+222886q^{22}+120894q^{21}-80040q^{20}-221925q^{19}-207773q^{18}-30735q^{17}+161460q^{16}+240301q^{15}+150885q^{14}-59247q^{13}-226308q^{12}-234155q^{11}-60229q^{10}+149756q^{9}+256463q^{8}+184503q^{7}-28946q^{6}-221404q^{5}-260122q^{4}-103337q^{3}+116818q^{2}+257165q+221314+25412q^{-1}-184231q^{-2}-265980q^{-3}-154497q^{-4}+49941q^{-5}+216685q^{-6}+235317q^{-7}+91357q^{-8}-105680q^{-9}-224132q^{-10}-180644q^{-11}-30045q^{-12}+131162q^{-13}+198022q^{-14}+129097q^{-15}-15560q^{-16}-137253q^{-17}-153672q^{-18}-78103q^{-19}+38491q^{-20}+118814q^{-21}+113070q^{-22}+38645q^{-23}-49053q^{-24}-89136q^{-25}-73297q^{-26}-15256q^{-27}+43494q^{-28}+64186q^{-29}+42586q^{-30}-315q^{-31}-31353q^{-32}-40384q^{-33}-23473q^{-34}+4683q^{-35}+22655q^{-36}+22982q^{-37}+9688q^{-38}-3983q^{-39}-13618q^{-40}-12778q^{-41}-3840q^{-42}+4221q^{-43}+7427q^{-44}+5249q^{-45}+1824q^{-46}-2636q^{-47}-4221q^{-48}-2246q^{-49}+28q^{-50}+1504q^{-51}+1457q^{-52}+1212q^{-53}-185q^{-54}-1038q^{-55}-619q^{-56}-191q^{-57}+202q^{-58}+215q^{-59}+411q^{-60}+51q^{-61}-239q^{-62}-102q^{-63}-52q^{-64}+27q^{-65}-11q^{-66}+110q^{-67}+22q^{-68}-60q^{-69}-4q^{-70}-8q^{-71}+11q^{-72}-19q^{-73}+23q^{-74}+7q^{-75}-16q^{-76}+4q^{-77}-2q^{-78}+4q^{-79}-4q^{-80}+2q^{-81}+2q^{-82}-3q^{-83}+q^{-84}$
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, 102]]
Out[2]=	PD[X[6, 2, 7, 1], X[16, 10, 17, 9], X[10, 3, 11, 4], X[2, 15, 3, 16], X[14, 5, 15, 6], X[18, 8, 19, 7], X[4, 11, 5, 12], X[8, 18, 9, 17], X[20, 14, 1, 13], X[12, 20, 13, 19]]
In[3]:=	GaussCode[Knot[10, 102]]
Out[3]=	GaussCode[1, -4, 3, -7, 5, -1, 6, -8, 2, -3, 7, -10, 9, -5, 4, -2, 8, -6, 10, -9]
In[4]:=	DTCode[Knot[10, 102]]
Out[4]=	DTCode[6, 10, 14, 18, 16, 4, 20, 2, 8, 12]
In[5]:=	br = BR[Knot[10, 102]]
Out[5]=	BR[4, {-1, -1, 2, -1, -3, 2, -1, 2, 2, 3, 3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 102]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 102]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[10, 102]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Chiral, 1, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 102]][t]
Out[10]=	2 8 16 2 3 21 - -- + -- - -- - 16 t + 8 t - 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 102]][z]
Out[11]=	2 4 6 1 - 2 z - 4 z - 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 102]}
In[13]:=	{KnotDet[Knot[10, 102]], KnotSignature[Knot[10, 102]]}
Out[13]=	{73, 0}
In[14]:=	Jones[Knot[10, 102]][q]
Out[14]=	-4 3 6 9 2 3 4 5 6 12 + q - -- + -- - - - 12 q + 11 q - 9 q + 6 q - 3 q + q 3 2 q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 102]}
In[16]:=	A2Invariant[Knot[10, 102]][q]
Out[16]=	-12 -10 -8 -6 2 3 2 6 8 10 -1 + q - q + q + q - -- + -- + q - 2 q + 2 q - 2 q + 4 2 q q 12 14 16 18 q + q - q + q
In[17]:=	HOMFLYPT[Knot[10, 102]][a, z]
Out[17]=	2 2 4 4 -4 -2 2 2 2 z 3 z 2 2 4 z 3 z a - a + a - 3 z + ---- - ---- + 2 a z - 3 z + -- - ---- + 4 2 4 2 a a a a 6 2 4 6 z a z - z - -- 2 a
In[18]:=	Kauffman[Knot[10, 102]][a, z]
Out[18]=	2 2 2 -4 -2 2 2 z 4 z 4 z 2 2 z 4 z 8 z a + a - a - --- - --- - --- - 2 a z + 2 z + ---- - ---- - ---- + 5 3 a 6 4 2 a a a a a 3 3 3 2 2 4 2 7 z 13 z 16 z 3 3 3 4 3 a z - a z + ---- + ----- + ----- + 7 a z - 3 a z + 3 z - 5 3 a a a 4 4 4 5 5 5 3 z 8 z 21 z 2 4 4 4 9 z 14 z 17 z ---- + ---- + ----- - 6 a z + a z - ---- - ----- - ----- - 6 4 2 5 3 a a a a a a 6 6 6 7 7 5 3 5 6 z 11 z 24 z 2 6 3 z z 9 a z + 3 a z - 7 z + -- - ----- - ----- + 5 a z + ---- + -- + 6 4 2 5 3 a a a a a 7 8 8 9 9 4 z 7 8 4 z 9 z 2 z 2 z ---- + 6 a z + 5 z + ---- + ---- + ---- + ---- a 4 2 3 a a a a
In[19]:=	{Vassiliev[2][Knot[10, 102]], Vassiliev[3][Knot[10, 102]]}
Out[19]=	{-2, -1}
In[20]:=	Kh[Knot[10, 102]][q, t]
Out[20]=	7 1 2 1 4 2 5 4 - + 6 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 6 q t + q 9 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 3 2 5 2 5 3 7 3 7 4 9 4 6 q t + 5 q t + 6 q t + 4 q t + 5 q t + 2 q t + 4 q t + 9 5 11 5 13 6 q t + 2 q t + q t
In[21]:=	ColouredJones[Knot[10, 102], 2][q]
Out[21]=	-12 3 2 6 15 10 18 45 24 46 87 28 83 + q - --- + --- + -- - -- + -- + -- - -- + -- + -- - -- + -- - 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q 2 3 4 5 6 7 8 113 q + 15 q + 104 q - 106 q - 7 q + 101 q - 74 q - 26 q + 9 10 11 12 13 14 15 16 76 q - 34 q - 29 q + 39 q - 6 q - 16 q + 10 q + q - 17 18 3 q + q

See/edit the Rolfsen_Splice_Template. Back to the top.

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