10 103

10_102

10_104

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10 103 Further Notes and Views

Knot presentations

Planar diagram presentation	X6271 X18,6,19,5 X20,13,1,14 X16,7,17,8 X10,3,11,4 X4,11,5,12 X14,9,15,10 X8,15,9,16 X12,19,13,20 X2,18,3,17
Gauss code	1, -10, 5, -6, 2, -1, 4, -8, 7, -5, 6, -9, 3, -7, 8, -4, 10, -2, 9, -3
Dowker-Thistlethwaite code	6 10 18 16 14 4 20 8 2 12
Conway Notation	[30:2:2]

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 3 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 2 Maximal Thurston-Bennequin number [-10][-2] Hyperbolic Volume 13.8748 A-Polynomial See Data:10 103/A-polynomial

[edit Notes for 10 103's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 10 103's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-8t^{2}+17t-21+17t^{-1}-8t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}+4z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{5,t+1\}$
Determinant and Signature	{ 75, -2 }
Jones polynomial	$-q^{2}+3q-6+10q^{-1}-11q^{-2}+13q^{-3}-12q^{-4}+9q^{-5}-6q^{-6}+3q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-z^{4}a^{6}-2z^{2}a^{6}-a^{6}+z^{6}a^{4}+3z^{4}a^{4}+3z^{2}a^{4}+z^{6}a^{2}+3z^{4}a^{2}+4z^{2}a^{2}+3a^{2}-z^{4}-2z^{2}-1$
Kauffman polynomial (db, data sources)	$z^{5}a^{9}-2z^{3}a^{9}+3z^{6}a^{8}-6z^{4}a^{8}+z^{2}a^{8}+5z^{7}a^{7}-12z^{5}a^{7}+10z^{3}a^{7}-4za^{7}+5z^{8}a^{6}-12z^{6}a^{6}+13z^{4}a^{6}-6z^{2}a^{6}+a^{6}+2z^{9}a^{5}+3z^{7}a^{5}-16z^{5}a^{5}+21z^{3}a^{5}-6za^{5}+9z^{8}a^{4}-23z^{6}a^{4}+25z^{4}a^{4}-8z^{2}a^{4}+2z^{9}a^{3}+2z^{7}a^{3}-9z^{5}a^{3}+9z^{3}a^{3}-2za^{3}+4z^{8}a^{2}-5z^{6}a^{2}+2z^{2}a^{2}-3a^{2}+4z^{7}a-5z^{5}a-2z^{3}a+za+3z^{6}-6z^{4}+3z^{2}-1+z^{5}a^{-1}-2z^{3}a^{-1}+za^{-1}$
The A2 invariant	$-q^{24}+q^{22}-q^{20}-q^{18}+2q^{16}-3q^{14}+q^{12}+q^{8}+4q^{6}-q^{4}+3q^{2}-1-q^{-2}+q^{-4}-q^{-6}$
The G2 invariant	$q^{128}-2q^{126}+4q^{124}-7q^{122}+7q^{120}-6q^{118}+12q^{114}-22q^{112}+35q^{110}-42q^{108}+35q^{106}-15q^{104}-25q^{102}+72q^{100}-108q^{98}+118q^{96}-88q^{94}+18q^{92}+74q^{90}-153q^{88}+184q^{86}-149q^{84}+53q^{82}+55q^{80}-142q^{78}+160q^{76}-106q^{74}+11q^{72}+92q^{70}-143q^{68}+115q^{66}-28q^{64}-92q^{62}+180q^{60}-204q^{58}+150q^{56}-38q^{54}-94q^{52}+208q^{50}-252q^{48}+218q^{46}-117q^{44}-25q^{42}+148q^{40}-206q^{38}+189q^{36}-98q^{34}-12q^{32}+112q^{30}-142q^{28}+98q^{26}-2q^{24}-100q^{22}+159q^{20}-140q^{18}+58q^{16}+52q^{14}-136q^{12}+176q^{10}-149q^{8}+80q^{6}+3q^{4}-78q^{2}+111-107q^{-2}+78q^{-4}-34q^{-6}-3q^{-8}+28q^{-10}-42q^{-12}+39q^{-14}-29q^{-16}+15q^{-18}-3q^{-20}-6q^{-22}+8q^{-24}-8q^{-26}+5q^{-28}-2q^{-30}+q^{-32}$

Further Quantum Invariants

Further quantum knot invariants for 10_103.

A1 Invariants.

Weight	Invariant
1	$-q^{17}+2q^{15}-3q^{13}+3q^{11}-3q^{9}+q^{7}+2q^{5}-q^{3}+4q-3q^{-1}+2q^{-3}-q^{-5}$
2	$q^{48}-2q^{46}+6q^{42}-8q^{40}-4q^{38}+19q^{36}-10q^{34}-19q^{32}+26q^{30}-27q^{26}+20q^{24}+11q^{22}-20q^{20}-q^{18}+13q^{16}-q^{14}-19q^{12}+13q^{10}+19q^{8}-26q^{6}+2q^{4}+27q^{2}-19-9q^{-2}+19q^{-4}-5q^{-6}-7q^{-8}+6q^{-10}-q^{-12}-2q^{-14}+q^{-16}$
3	$-q^{93}+2q^{91}-3q^{87}+7q^{83}+q^{81}-18q^{79}-5q^{77}+34q^{75}+22q^{73}-47q^{71}-58q^{69}+48q^{67}+103q^{65}-24q^{63}-143q^{61}-25q^{59}+165q^{57}+82q^{55}-158q^{53}-132q^{51}+125q^{49}+164q^{47}-83q^{45}-170q^{43}+36q^{41}+153q^{39}+16q^{37}-131q^{35}-50q^{33}+94q^{31}+91q^{29}-64q^{27}-128q^{25}+20q^{23}+159q^{21}+25q^{19}-175q^{17}-73q^{15}+168q^{13}+124q^{11}-134q^{9}-151q^{7}+82q^{5}+160q^{3}-29q-131q^{-1}-19q^{-3}+95q^{-5}+37q^{-7}-53q^{-9}-34q^{-11}+22q^{-13}+22q^{-15}-8q^{-17}-12q^{-19}+5q^{-21}+4q^{-23}-3q^{-25}-2q^{-27}+q^{-29}+2q^{-31}-q^{-33}$
4	$q^{152}-2q^{150}+3q^{146}-3q^{144}+q^{142}-5q^{140}+7q^{138}+16q^{136}-17q^{134}-17q^{132}-28q^{130}+31q^{128}+93q^{126}+6q^{124}-76q^{122}-177q^{120}-34q^{118}+246q^{116}+248q^{114}+47q^{112}-424q^{110}-459q^{108}+95q^{106}+597q^{104}+677q^{102}-224q^{100}-974q^{98}-687q^{96}+363q^{94}+1355q^{92}+666q^{90}-793q^{88}-1447q^{86}-557q^{84}+1248q^{82}+1443q^{80}+59q^{78}-1390q^{76}-1282q^{74}+494q^{72}+1410q^{70}+729q^{68}-738q^{66}-1277q^{64}-178q^{62}+874q^{60}+884q^{58}-132q^{56}-910q^{54}-543q^{52}+378q^{50}+870q^{48}+316q^{46}-600q^{44}-909q^{42}-79q^{40}+938q^{38}+866q^{36}-236q^{34}-1315q^{32}-732q^{30}+778q^{28}+1434q^{26}+479q^{24}-1294q^{22}-1398q^{20}+57q^{18}+1430q^{16}+1246q^{14}-530q^{12}-1410q^{10}-784q^{8}+623q^{6}+1304q^{4}+351q^{2}-631-903q^{-2}-218q^{-4}+619q^{-6}+523q^{-8}+85q^{-10}-394q^{-12}-359q^{-14}+53q^{-16}+187q^{-18}+189q^{-20}-26q^{-22}-128q^{-24}-41q^{-26}-7q^{-28}+58q^{-30}+17q^{-32}-16q^{-34}+2q^{-36}-13q^{-38}+6q^{-40}-q^{-42}-4q^{-44}+6q^{-46}-q^{-48}+2q^{-50}-q^{-52}-2q^{-54}+q^{-56}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{24}+q^{22}-q^{20}-q^{18}+2q^{16}-3q^{14}+q^{12}+q^{8}+4q^{6}-q^{4}+3q^{2}-1-q^{-2}+q^{-4}-q^{-6}$
1,1	$q^{68}-4q^{66}+10q^{64}-22q^{62}+46q^{60}-78q^{58}+124q^{56}-202q^{54}+301q^{52}-406q^{50}+528q^{48}-644q^{46}+711q^{44}-702q^{42}+606q^{40}-428q^{38}+145q^{36}+198q^{34}-556q^{32}+896q^{30}-1167q^{28}+1354q^{26}-1420q^{24}+1360q^{22}-1198q^{20}+912q^{18}-578q^{16}+228q^{14}+115q^{12}-388q^{10}+596q^{8}-672q^{6}+682q^{4}-632q^{2}+534-418q^{-2}+311q^{-4}-224q^{-6}+146q^{-8}-92q^{-10}+54q^{-12}-28q^{-14}+12q^{-16}-4q^{-18}+q^{-20}$
2,0	$q^{62}-q^{60}+2q^{56}-q^{54}-2q^{52}-q^{50}+7q^{48}+q^{46}-11q^{44}+9q^{40}-q^{38}-13q^{36}+2q^{34}+12q^{32}-2q^{30}-10q^{28}+5q^{26}+3q^{24}-11q^{22}+3q^{20}+5q^{18}-3q^{16}+12q^{12}+3q^{10}-10q^{8}+4q^{6}+14q^{4}-6q^{2}-11+7q^{-2}+7q^{-4}-5q^{-6}-6q^{-8}+2q^{-10}+3q^{-12}-2q^{-14}-q^{-16}+q^{-18}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{54}-2q^{52}+4q^{48}-6q^{46}+2q^{44}+11q^{42}-15q^{40}+2q^{38}+17q^{36}-22q^{34}-q^{32}+17q^{30}-16q^{28}-4q^{26}+10q^{24}-3q^{22}-6q^{20}-q^{18}+12q^{16}-10q^{12}+22q^{10}+5q^{8}-20q^{6}+18q^{4}+2q^{2}-17+9q^{-2}+q^{-4}-8q^{-6}+4q^{-8}+q^{-10}-2q^{-12}+q^{-14}$
1,0,0	$-q^{31}+q^{29}-2q^{27}+q^{25}-2q^{23}+2q^{21}-3q^{19}+q^{17}-q^{15}+q^{13}+2q^{11}+2q^{9}+4q^{7}-q^{5}+3q^{3}-2q+q^{-1}-2q^{-3}+q^{-5}-q^{-7}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{68}-q^{66}-2q^{64}+4q^{62}+q^{60}-8q^{58}+4q^{56}+11q^{54}-6q^{52}-8q^{50}+12q^{48}+9q^{46}-18q^{44}-10q^{42}+14q^{40}-6q^{38}-22q^{36}+8q^{34}+10q^{32}-18q^{30}-2q^{28}+15q^{26}-3q^{24}-9q^{22}+17q^{20}+19q^{18}-9q^{16}+2q^{14}+22q^{12}+q^{10}-16q^{8}+5q^{6}+7q^{4}-8q^{2}-8+2q^{-2}+2q^{-4}-4q^{-6}-q^{-8}+3q^{-10}-q^{-14}+q^{-16}$
1,0,0,0	$-q^{38}+q^{36}-2q^{34}-2q^{28}+2q^{26}-3q^{24}+q^{22}-q^{20}+q^{16}+2q^{14}+3q^{12}+2q^{10}+4q^{8}-q^{6}+3q^{4}-2q^{2}-2q^{-4}+q^{-6}-q^{-8}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{54}+2q^{52}-4q^{50}+8q^{48}-12q^{46}+16q^{44}-23q^{42}+25q^{40}-26q^{38}+23q^{36}-16q^{34}+9q^{32}+3q^{30}-16q^{28}+30q^{26}-42q^{24}+47q^{22}-50q^{20}+47q^{18}-42q^{16}+32q^{14}-16q^{12}+6q^{10}+9q^{8}-14q^{6}+24q^{4}-26q^{2}+25-23q^{-2}+17q^{-4}-12q^{-6}+8q^{-8}-5q^{-10}+2q^{-12}-q^{-14}$
1,0	$q^{88}-2q^{84}-2q^{82}+2q^{80}+6q^{78}-9q^{74}-6q^{72}+9q^{70}+17q^{68}-3q^{66}-23q^{64}-10q^{62}+21q^{60}+22q^{58}-12q^{56}-28q^{54}-3q^{52}+25q^{50}+12q^{48}-19q^{46}-17q^{44}+11q^{42}+17q^{40}-7q^{38}-18q^{36}+q^{34}+17q^{32}+2q^{30}-17q^{28}-6q^{26}+18q^{24}+14q^{22}-14q^{20}-16q^{18}+13q^{16}+28q^{14}-q^{12}-26q^{10}-12q^{8}+23q^{6}+21q^{4}-10q^{2}-23-4q^{-2}+16q^{-4}+9q^{-6}-7q^{-8}-10q^{-10}-q^{-12}+6q^{-14}+3q^{-16}-2q^{-18}-2q^{-20}+q^{-24}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{74}-2q^{72}+2q^{70}-4q^{68}+7q^{66}-9q^{64}+11q^{62}-13q^{60}+19q^{58}-20q^{56}+20q^{54}-19q^{52}+19q^{50}-17q^{48}+6q^{46}-7q^{44}-q^{42}+8q^{40}-21q^{38}+22q^{36}-29q^{34}+37q^{32}-38q^{30}+36q^{28}-38q^{26}+38q^{24}-27q^{22}+26q^{20}-17q^{18}+15q^{16}+4q^{14}+10q^{10}-14q^{8}+20q^{6}-20q^{4}+18q^{2}-22+17q^{-2}-15q^{-4}+10q^{-6}-10q^{-8}+7q^{-10}-4q^{-12}+3q^{-14}-2q^{-16}+q^{-18}$

G2 Invariants.

Weight

Invariant

1,0

$q^{128}-2q^{126}+4q^{124}-7q^{122}+7q^{120}-6q^{118}+12q^{114}-22q^{112}+35q^{110}-42q^{108}+35q^{106}-15q^{104}-25q^{102}+72q^{100}-108q^{98}+118q^{96}-88q^{94}+18q^{92}+74q^{90}-153q^{88}+184q^{86}-149q^{84}+53q^{82}+55q^{80}-142q^{78}+160q^{76}-106q^{74}+11q^{72}+92q^{70}-143q^{68}+115q^{66}-28q^{64}-92q^{62}+180q^{60}-204q^{58}+150q^{56}-38q^{54}-94q^{52}+208q^{50}-252q^{48}+218q^{46}-117q^{44}-25q^{42}+148q^{40}-206q^{38}+189q^{36}-98q^{34}-12q^{32}+112q^{30}-142q^{28}+98q^{26}-2q^{24}-100q^{22}+159q^{20}-140q^{18}+58q^{16}+52q^{14}-136q^{12}+176q^{10}-149q^{8}+80q^{6}+3q^{4}-78q^{2}+111-107q^{-2}+78q^{-4}-34q^{-6}-3q^{-8}+28q^{-10}-42q^{-12}+39q^{-14}-29q^{-16}+15q^{-18}-3q^{-20}-6q^{-22}+8q^{-24}-8q^{-26}+5q^{-28}-2q^{-30}+q^{-32}$

.

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["10 103"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-8t^{2}+17t-21+17t^{-1}-8t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

$\{5,t+1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 75, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{2}+3q-6+10q^{-1}-11q^{-2}+13q^{-3}-12q^{-4}+9q^{-5}-6q^{-6}+3q^{-7}-q^{-8}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{5}a^{9}-2z^{3}a^{9}+3z^{6}a^{8}-6z^{4}a^{8}+z^{2}a^{8}+5z^{7}a^{7}-12z^{5}a^{7}+10z^{3}a^{7}-4za^{7}+5z^{8}a^{6}-12z^{6}a^{6}+13z^{4}a^{6}-6z^{2}a^{6}+a^{6}+2z^{9}a^{5}+3z^{7}a^{5}-16z^{5}a^{5}+21z^{3}a^{5}-6za^{5}+9z^{8}a^{4}-23z^{6}a^{4}+25z^{4}a^{4}-8z^{2}a^{4}+2z^{9}a^{3}+2z^{7}a^{3}-9z^{5}a^{3}+9z^{3}a^{3}-2za^{3}+4z^{8}a^{2}-5z^{6}a^{2}+2z^{2}a^{2}-3a^{2}+4z^{7}a-5z^{5}a-2z^{3}a+za+3z^{6}-6z^{4}+3z^{2}-1+z^{5}a^{-1}-2z^{3}a^{-1}+za^{-1}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_40, ...}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {10_40, ...}

Vassiliev invariants

V2 and V3:

(3, -4)

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 $12$ $-32$ $72$ $126$ $2$ $-384$ $-{\frac {1760}{3}}$ $-{\frac {128}{3}}$ $-96$ $288$ $512$ $1512$ $24$ ${\frac {27311}{10}}$ ${\frac {454}{15}}$ ${\frac {4234}{5}}$ ${\frac {155}{2}}$ ${\frac {431}{10}}$

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 10 103. 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                   2   2 1                 4 1   -3 -1               6 2     4 -3             6 5       -1 -5           7 5         2 -7         5 6           1 -9       4 7             -3 -11     2 5               3 -13   1 4                 -3 -15   2                   2 -17 1                     -1

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{7}-3q^{6}+q^{5}+8q^{4}-16q^{3}+3q^{2}+32q-44-7q^{-1}+78q^{-2}-69q^{-3}-35q^{-4}+123q^{-5}-75q^{-6}-67q^{-7}+141q^{-8}-61q^{-9}-81q^{-10}+122q^{-11}-30q^{-12}-72q^{-13}+75q^{-14}-3q^{-15}-46q^{-16}+30q^{-17}+6q^{-18}-17q^{-19}+7q^{-20}+2q^{-21}-3q^{-22}+q^{-23}$
3	$-q^{15}+3q^{14}-q^{13}-3q^{12}-2q^{11}+10q^{10}-20q^{8}+2q^{7}+40q^{6}-76q^{4}-17q^{3}+130q^{2}+58q-190-129q^{-1}+232q^{-2}+247q^{-3}-268q^{-4}-362q^{-5}+249q^{-6}+505q^{-7}-224q^{-8}-603q^{-9}+147q^{-10}+705q^{-11}-90q^{-12}-742q^{-13}-q^{-14}+769q^{-15}+65q^{-16}-739q^{-17}-145q^{-18}+688q^{-19}+212q^{-20}-602q^{-21}-262q^{-22}+482q^{-23}+299q^{-24}-355q^{-25}-301q^{-26}+225q^{-27}+273q^{-28}-115q^{-29}-218q^{-30}+35q^{-31}+155q^{-32}+4q^{-33}-91q^{-34}-20q^{-35}+49q^{-36}+15q^{-37}-22q^{-38}-8q^{-39}+10q^{-40}+2q^{-41}-3q^{-42}-2q^{-43}+3q^{-44}-q^{-45}$
4	$q^{26}-3q^{25}+q^{24}+3q^{23}-3q^{22}+8q^{21}-13q^{20}+4q^{19}+10q^{18}-22q^{17}+23q^{16}-31q^{15}+37q^{14}+51q^{13}-87q^{12}-11q^{11}-118q^{10}+139q^{9}+266q^{8}-89q^{7}-145q^{6}-530q^{5}+104q^{4}+745q^{3}+349q^{2}-49q-1367-581q^{-1}+1017q^{-2}+1331q^{-3}+904q^{-4}-2048q^{-5}-1988q^{-6}+391q^{-7}+2211q^{-8}+2680q^{-9}-1864q^{-10}-3361q^{-11}-1064q^{-12}+2315q^{-13}+4453q^{-14}-909q^{-15}-4017q^{-16}-2574q^{-17}+1732q^{-18}+5532q^{-19}+193q^{-20}-3945q^{-21}-3591q^{-22}+902q^{-23}+5841q^{-24}+1109q^{-25}-3391q^{-26}-4083q^{-27}-19q^{-28}+5474q^{-29}+1887q^{-30}-2375q^{-31}-4093q^{-32}-1071q^{-33}+4375q^{-34}+2426q^{-35}-908q^{-36}-3412q^{-37}-1987q^{-38}+2599q^{-39}+2318q^{-40}+541q^{-41}-2028q^{-42}-2182q^{-43}+794q^{-44}+1428q^{-45}+1195q^{-46}-569q^{-47}-1493q^{-48}-198q^{-49}+378q^{-50}+908q^{-51}+181q^{-52}-592q^{-53}-278q^{-54}-124q^{-55}+354q^{-56}+216q^{-57}-121q^{-58}-77q^{-59}-126q^{-60}+74q^{-61}+73q^{-62}-20q^{-63}+5q^{-64}-39q^{-65}+12q^{-66}+14q^{-67}-9q^{-68}+5q^{-69}-6q^{-70}+3q^{-71}+2q^{-72}-3q^{-73}+q^{-74}$
5	$-q^{40}+3q^{39}-q^{38}-3q^{37}+3q^{36}-3q^{35}-5q^{34}+9q^{33}+6q^{32}-6q^{31}+11q^{30}-5q^{29}-31q^{28}-12q^{27}+8q^{26}+27q^{25}+72q^{24}+56q^{23}-65q^{22}-174q^{21}-161q^{20}-q^{19}+298q^{18}+459q^{17}+219q^{16}-385q^{15}-899q^{14}-761q^{13}+192q^{12}+1400q^{11}+1756q^{10}+564q^{9}-1671q^{8}-3127q^{7}-2139q^{6}+1184q^{5}+4518q^{4}+4662q^{3}+529q^{2}-5289q-7834-3775q^{-1}+4744q^{-2}+10896q^{-3}+8472q^{-4}-2221q^{-5}-13154q^{-6}-14015q^{-7}-2144q^{-8}+13593q^{-9}+19434q^{-10}+8337q^{-11}-12153q^{-12}-23999q^{-13}-15026q^{-14}+8685q^{-15}+26814q^{-16}+21869q^{-17}-4042q^{-18}-28061q^{-19}-27448q^{-20}-1273q^{-21}+27591q^{-22}+31996q^{-23}+6247q^{-24}-26227q^{-25}-34797q^{-26}-10731q^{-27}+24132q^{-28}+36701q^{-29}+14253q^{-30}-21994q^{-31}-37381q^{-32}-17209q^{-33}+19591q^{-34}+37727q^{-35}+19581q^{-36}-17184q^{-37}-37280q^{-38}-21810q^{-39}+14232q^{-40}+36397q^{-41}+23911q^{-42}-10754q^{-43}-34576q^{-44}-25775q^{-45}+6436q^{-46}+31610q^{-47}+27167q^{-48}-1485q^{-49}-27347q^{-50}-27508q^{-51}-3652q^{-52}+21653q^{-53}+26442q^{-54}+8385q^{-55}-15075q^{-56}-23661q^{-57}-11847q^{-58}+8232q^{-59}+19270q^{-60}+13533q^{-61}-2099q^{-62}-13857q^{-63}-13171q^{-64}-2526q^{-65}+8308q^{-66}+11063q^{-67}+5126q^{-68}-3476q^{-69}-7946q^{-70}-5816q^{-71}+105q^{-72}+4713q^{-73}+4988q^{-74}+1705q^{-75}-2036q^{-76}-3525q^{-77}-2172q^{-78}+373q^{-79}+1989q^{-80}+1817q^{-81}+431q^{-82}-877q^{-83}-1195q^{-84}-581q^{-85}+241q^{-86}+640q^{-87}+446q^{-88}+5q^{-89}-261q^{-90}-266q^{-91}-77q^{-92}+109q^{-93}+125q^{-94}+36q^{-95}-21q^{-96}-41q^{-97}-37q^{-98}+12q^{-99}+25q^{-100}-2q^{-101}-3q^{-102}+3q^{-103}-6q^{-104}-q^{-105}+6q^{-106}-3q^{-107}-2q^{-108}+3q^{-109}-q^{-110}$
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, 103]]
Out[2]=	PD[X[6, 2, 7, 1], X[18, 6, 19, 5], X[20, 13, 1, 14], X[16, 7, 17, 8], X[10, 3, 11, 4], X[4, 11, 5, 12], X[14, 9, 15, 10], X[8, 15, 9, 16], X[12, 19, 13, 20], X[2, 18, 3, 17]]
In[3]:=	GaussCode[Knot[10, 103]]
Out[3]=	GaussCode[1, -10, 5, -6, 2, -1, 4, -8, 7, -5, 6, -9, 3, -7, 8, -4, 10, -2, 9, -3]
In[4]:=	DTCode[Knot[10, 103]]
Out[4]=	DTCode[6, 10, 18, 16, 14, 4, 20, 8, 2, 12]
In[5]:=	br = BR[Knot[10, 103]]
Out[5]=	BR[4, {-1, -1, -2, 1, 3, -2, -2, 3, -2, -2, 3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 103]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 103]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[10, 103]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 3, 3, 3, NotAvailable, 2}
In[10]:=	alex = Alexander[Knot[10, 103]][t]
Out[10]=	2 8 17 2 3 -21 + -- - -- + -- + 17 t - 8 t + 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 103]][z]
Out[11]=	2 4 6 1 + 3 z + 4 z + 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 40], Knot[10, 103]}
In[13]:=	{KnotDet[Knot[10, 103]], KnotSignature[Knot[10, 103]]}
Out[13]=	{75, -2}
In[14]:=	Jones[Knot[10, 103]][q]
Out[14]=	-8 3 6 9 12 13 11 10 2 -6 - q + -- - -- + -- - -- + -- - -- + -- + 3 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, 40], Knot[10, 103]}
In[16]:=	A2Invariant[Knot[10, 103]][q]
Out[16]=	-24 -22 -20 -18 2 3 -12 -8 4 -4 -1 - q + q - q - q + --- - --- + q + q + -- - q + 16 14 6 q q q 3 2 4 6 -- - q + q - q 2 q
In[17]:=	HOMFLYPT[Knot[10, 103]][a, z]
Out[17]=	2 6 2 2 2 4 2 6 2 4 2 4 -1 + 3 a - a - 2 z + 4 a z + 3 a z - 2 a z - z + 3 a z + 4 4 6 4 2 6 4 6 3 a z - a z + a z + a z
In[18]:=	Kauffman[Knot[10, 103]][a, z]
Out[18]=	2 6 z 3 5 7 2 2 2 -1 - 3 a + a + - + a z - 2 a z - 6 a z - 4 a z + 3 z + 2 a z - a 3 4 2 6 2 8 2 2 z 3 3 3 5 3 8 a z - 6 a z + a z - ---- - 2 a z + 9 a z + 21 a z + a 5 7 3 9 3 4 4 4 6 4 8 4 z 10 a z - 2 a z - 6 z + 25 a z + 13 a z - 6 a z + -- - a 5 3 5 5 5 7 5 9 5 6 2 6 5 a z - 9 a z - 16 a z - 12 a z + a z + 3 z - 5 a z - 4 6 6 6 8 6 7 3 7 5 7 23 a z - 12 a z + 3 a z + 4 a z + 2 a z + 3 a z + 7 7 2 8 4 8 6 8 3 9 5 9 5 a z + 4 a z + 9 a z + 5 a z + 2 a z + 2 a z
In[19]:=	{Vassiliev[2][Knot[10, 103]], Vassiliev[3][Knot[10, 103]]}
Out[19]=	{3, -4}
In[20]:=	Kh[Knot[10, 103]][q, t]
Out[20]=	5 6 1 2 1 4 2 5 4 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 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 7 5 6 7 5 6 2 t 2 ----- + ----- + ----- + ----- + ---- + ---- + --- + 4 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 2 q t + q t
In[21]:=	ColouredJones[Knot[10, 103], 2][q]
Out[21]=	-23 3 2 7 17 6 30 46 3 75 -44 + q - --- + --- + --- - --- + --- + --- - --- - --- + --- - 22 21 20 19 18 17 16 15 14 q q q q q q q q q 72 30 122 81 61 141 67 75 123 35 69 78 7 --- - --- + --- - --- - -- + --- - -- - -- + --- - -- - -- + -- - - + 13 12 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q q q 2 3 4 5 6 7 32 q + 3 q - 16 q + 8 q + q - 3 q + q

See/edit the Rolfsen_Splice_Template.