10 113

10_112

10_114

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10 113 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X10,4,11,3 X14,6,15,5 X20,16,1,15 X12,7,13,8 X8,18,9,17 X6,19,7,20 X16,12,17,11 X18,13,19,14 X2,10,3,9
Gauss code	1, -10, 2, -1, 3, -7, 5, -6, 10, -2, 8, -5, 9, -3, 4, -8, 6, -9, 7, -4
Dowker-Thistlethwaite code	4 10 14 12 2 16 18 20 8 6
Conway Notation	[8*21]

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 1 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-2][-10] Hyperbolic Volume 16.4735 A-Polynomial See Data:10 113/A-polynomial

[edit Notes for 10 113'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 10 113's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-11t^{2}+26t-33+26t^{-1}-11t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}+z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 111, 2 }
Jones polynomial	$-q^{8}+5q^{7}-10q^{6}+14q^{5}-18q^{4}+19q^{3}-17q^{2}+14q-8+4q^{-1}-q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+z^{6}a^{-4}+2z^{4}a^{-2}+z^{4}a^{-4}-z^{4}a^{-6}-z^{4}+3z^{2}a^{-2}-2z^{2}a^{-4}-z^{2}+3a^{-2}-3a^{-4}+a^{-6}$
Kauffman polynomial (db, data sources)	$3z^{9}a^{-3}+3z^{9}a^{-5}+7z^{8}a^{-2}+16z^{8}a^{-4}+9z^{8}a^{-6}+7z^{7}a^{-1}+12z^{7}a^{-3}+15z^{7}a^{-5}+10z^{7}a^{-7}-5z^{6}a^{-2}-23z^{6}a^{-4}-9z^{6}a^{-6}+5z^{6}a^{-8}+4z^{6}+az^{5}-10z^{5}a^{-1}-30z^{5}a^{-3}-36z^{5}a^{-5}-16z^{5}a^{-7}+z^{5}a^{-9}-6z^{4}a^{-2}+z^{4}a^{-4}-4z^{4}a^{-6}-5z^{4}a^{-8}-6z^{4}-az^{3}+5z^{3}a^{-1}+17z^{3}a^{-3}+16z^{3}a^{-5}+5z^{3}a^{-7}+8z^{2}a^{-2}+8z^{2}a^{-4}+3z^{2}a^{-6}+3z^{2}-za^{-1}-za^{-3}+za^{-5}+za^{-7}-3a^{-2}-3a^{-4}-a^{-6}$
The A2 invariant	$-q^{6}+2q^{4}-q^{2}-1+5q^{-2}-2q^{-4}+4q^{-6}-2q^{-10}+q^{-12}-5q^{-14}+3q^{-16}-q^{-18}-q^{-20}+3q^{-22}-q^{-24}$
The G2 invariant	$q^{32}-3q^{30}+7q^{28}-13q^{26}+15q^{24}-15q^{22}+4q^{20}+20q^{18}-50q^{16}+86q^{14}-108q^{12}+100q^{10}-52q^{8}-47q^{6}+182q^{4}-301q^{2}+359-303q^{-2}+110q^{-4}+168q^{-6}-443q^{-8}+605q^{-10}-562q^{-12}+315q^{-14}+63q^{-16}-415q^{-18}+597q^{-20}-516q^{-22}+220q^{-24}+165q^{-26}-445q^{-28}+484q^{-30}-265q^{-32}-118q^{-34}+506q^{-36}-702q^{-38}+623q^{-40}-277q^{-42}-216q^{-44}+661q^{-46}-905q^{-48}+841q^{-50}-511q^{-52}+19q^{-54}+454q^{-56}-748q^{-58}+762q^{-60}-503q^{-62}+81q^{-64}+313q^{-66}-530q^{-68}+465q^{-70}-168q^{-72}-207q^{-74}+498q^{-76}-548q^{-78}+346q^{-80}+29q^{-82}-415q^{-84}+644q^{-86}-631q^{-88}+398q^{-90}-50q^{-92}-270q^{-94}+454q^{-96}-460q^{-98}+334q^{-100}-139q^{-102}-42q^{-104}+147q^{-106}-183q^{-108}+147q^{-110}-84q^{-112}+31q^{-114}+10q^{-116}-25q^{-118}+26q^{-120}-20q^{-122}+10q^{-124}-4q^{-126}+q^{-128}$

Further Quantum Invariants

Further quantum knot invariants for 10_113.

A1 Invariants.

Weight	Invariant
1	$-q^{5}+3q^{3}-4q+6q^{-1}-3q^{-3}+2q^{-5}+q^{-7}-4q^{-9}+4q^{-11}-5q^{-13}+4q^{-15}-q^{-17}$
2	$q^{16}-3q^{14}+10q^{10}-15q^{8}-6q^{6}+39q^{4}-26q^{2}-34+67q^{-2}-10q^{-4}-61q^{-6}+51q^{-8}+19q^{-10}-50q^{-12}+6q^{-14}+35q^{-16}-10q^{-18}-40q^{-20}+32q^{-22}+34q^{-24}-65q^{-26}+10q^{-28}+61q^{-30}-52q^{-32}-17q^{-34}+50q^{-36}-17q^{-38}-20q^{-40}+16q^{-42}+q^{-44}-4q^{-46}+q^{-48}$
3	$-q^{33}+3q^{31}-6q^{27}-q^{25}+15q^{23}+5q^{21}-43q^{19}-14q^{17}+84q^{15}+58q^{13}-140q^{11}-150q^{9}+181q^{7}+296q^{5}-167q^{3}-458q+67q^{-1}+608q^{-3}+93q^{-5}-663q^{-7}-290q^{-9}+618q^{-11}+460q^{-13}-492q^{-15}-556q^{-17}+304q^{-19}+579q^{-21}-105q^{-23}-535q^{-25}-86q^{-27}+459q^{-29}+252q^{-31}-346q^{-33}-413q^{-35}+228q^{-37}+541q^{-39}-78q^{-41}-643q^{-43}-92q^{-45}+678q^{-47}+275q^{-49}-626q^{-51}-442q^{-53}+498q^{-55}+535q^{-57}-304q^{-59}-540q^{-61}+107q^{-63}+454q^{-65}+41q^{-67}-318q^{-69}-104q^{-71}+171q^{-73}+108q^{-75}-69q^{-77}-75q^{-79}+22q^{-81}+31q^{-83}-12q^{-87}-q^{-89}+4q^{-91}-q^{-93}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{6}+2q^{4}-q^{2}-1+5q^{-2}-2q^{-4}+4q^{-6}-2q^{-10}+q^{-12}-5q^{-14}+3q^{-16}-q^{-18}-q^{-20}+3q^{-22}-q^{-24}$
1,1	$q^{20}-6q^{18}+20q^{16}-50q^{14}+109q^{12}-218q^{10}+392q^{8}-662q^{6}+1048q^{4}-1548q^{2}+2134-2722q^{-2}+3230q^{-4}-3482q^{-6}+3384q^{-8}-2820q^{-10}+1769q^{-12}-308q^{-14}-1436q^{-16}+3228q^{-18}-4886q^{-20}+6174q^{-22}-6924q^{-24}+7056q^{-26}-6543q^{-28}+5466q^{-30}-3938q^{-32}+2154q^{-34}-340q^{-36}-1288q^{-38}+2552q^{-40}-3344q^{-42}+3642q^{-44}-3508q^{-46}+3056q^{-48}-2430q^{-50}+1773q^{-52}-1192q^{-54}+730q^{-56}-400q^{-58}+198q^{-60}-88q^{-62}+32q^{-64}-8q^{-66}+q^{-68}$
2,0	$q^{18}-2q^{16}-2q^{14}+6q^{12}+2q^{10}-12q^{8}-6q^{6}+19q^{4}+10q^{2}-24-7q^{-2}+35q^{-4}+7q^{-6}-29q^{-8}+4q^{-10}+24q^{-12}-7q^{-14}-22q^{-16}+7q^{-18}+6q^{-20}-20q^{-22}+8q^{-24}+13q^{-26}-15q^{-28}-4q^{-30}+28q^{-32}+2q^{-34}-30q^{-36}+4q^{-38}+27q^{-40}-3q^{-42}-29q^{-44}+7q^{-46}+19q^{-48}-6q^{-50}-11q^{-52}+9q^{-56}-q^{-58}-3q^{-60}+q^{-62}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{14}-3q^{12}+q^{10}+7q^{8}-14q^{6}+3q^{4}+24q^{2}-31+2q^{-2}+44q^{-4}-44q^{-6}+q^{-8}+48q^{-10}-34q^{-12}-7q^{-14}+27q^{-16}-10q^{-18}-17q^{-20}-5q^{-22}+20q^{-24}-6q^{-26}-31q^{-28}+41q^{-30}+9q^{-32}-48q^{-34}+39q^{-36}+9q^{-38}-42q^{-40}+26q^{-42}+5q^{-44}-19q^{-46}+11q^{-48}+2q^{-50}-4q^{-52}+q^{-54}$
1,0,0	$-q^{7}+2q^{5}-2q^{3}+2q-2q^{-1}+5q^{-3}-2q^{-5}+5q^{-7}+q^{-9}+q^{-11}-q^{-13}-3q^{-15}-5q^{-19}+3q^{-21}-2q^{-23}+3q^{-25}-2q^{-27}+3q^{-29}-q^{-31}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{16}-2q^{14}-q^{12}+5q^{10}-2q^{8}-9q^{6}+5q^{4}+14q^{2}-10-16q^{-2}+22q^{-4}+22q^{-6}-31q^{-8}-8q^{-10}+47q^{-12}+2q^{-14}-40q^{-16}+18q^{-18}+33q^{-20}-32q^{-22}-27q^{-24}+30q^{-26}-4q^{-28}-45q^{-30}+20q^{-32}+34q^{-34}-35q^{-36}-8q^{-38}+51q^{-40}-2q^{-42}-40q^{-44}+13q^{-46}+30q^{-48}-22q^{-50}-24q^{-52}+21q^{-54}+12q^{-56}-17q^{-58}-2q^{-60}+11q^{-62}-2q^{-64}-3q^{-66}+q^{-68}$
1,0,0,0	$-q^{8}+2q^{6}-2q^{4}+q^{2}+1-2q^{-2}+5q^{-4}-2q^{-6}+5q^{-8}+2q^{-10}+2q^{-12}+q^{-14}-q^{-16}-2q^{-18}-4q^{-20}-5q^{-24}+3q^{-26}-2q^{-28}+2q^{-30}+2q^{-32}-2q^{-34}+3q^{-36}-q^{-38}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{14}+3q^{12}-7q^{10}+13q^{8}-22q^{6}+33q^{4}-44q^{2}+55-58q^{-2}+58q^{-4}-46q^{-6}+29q^{-8}-2q^{-10}-26q^{-12}+57q^{-14}-83q^{-16}+104q^{-18}-115q^{-20}+113q^{-22}-102q^{-24}+78q^{-26}-51q^{-28}+19q^{-30}+9q^{-32}-34q^{-34}+51q^{-36}-59q^{-38}+60q^{-40}-54q^{-42}+45q^{-44}-31q^{-46}+19q^{-48}-10q^{-50}+4q^{-52}-q^{-54}$
1,0	$q^{24}-3q^{20}-3q^{18}+4q^{16}+10q^{14}-18q^{10}-14q^{8}+18q^{6}+34q^{4}-q^{2}-47-25q^{-2}+42q^{-4}+54q^{-6}-17q^{-8}-64q^{-10}-11q^{-12}+60q^{-14}+35q^{-16}-39q^{-18}-44q^{-20}+22q^{-22}+45q^{-24}-8q^{-26}-45q^{-28}-4q^{-30}+39q^{-32}+9q^{-34}-39q^{-36}-22q^{-38}+35q^{-40}+32q^{-42}-30q^{-44}-45q^{-46}+21q^{-48}+60q^{-50}+5q^{-52}-61q^{-54}-32q^{-56}+48q^{-58}+54q^{-60}-21q^{-62}-57q^{-64}-10q^{-66}+40q^{-68}+26q^{-70}-19q^{-72}-25q^{-74}+q^{-76}+15q^{-78}+6q^{-80}-4q^{-82}-4q^{-84}+q^{-88}$

D4 Invariants.

Weight

Invariant

1,0,0,0

$q^{18}-3q^{16}+4q^{14}-6q^{12}+11q^{10}-18q^{8}+21q^{6}-25q^{4}+37q^{2}-43+43q^{-2}-43q^{-4}+50q^{-6}-40q^{-8}+29q^{-10}-17q^{-12}+12q^{-14}+18q^{-16}-32q^{-18}+43q^{-20}-59q^{-22}+77q^{-24}-88q^{-26}+79q^{-28}-93q^{-30}+85q^{-32}-74q^{-34}+60q^{-36}-52q^{-38}+35q^{-40}-6q^{-42}-q^{-44}+14q^{-46}-30q^{-48}+44q^{-50}-45q^{-52}+45q^{-54}-50q^{-56}+44q^{-58}-35q^{-60}+30q^{-62}-25q^{-64}+16q^{-66}-8q^{-68}+6q^{-70}-4q^{-72}+q^{-74}$

G2 Invariants.

Weight

Invariant

1,0

$q^{32}-3q^{30}+7q^{28}-13q^{26}+15q^{24}-15q^{22}+4q^{20}+20q^{18}-50q^{16}+86q^{14}-108q^{12}+100q^{10}-52q^{8}-47q^{6}+182q^{4}-301q^{2}+359-303q^{-2}+110q^{-4}+168q^{-6}-443q^{-8}+605q^{-10}-562q^{-12}+315q^{-14}+63q^{-16}-415q^{-18}+597q^{-20}-516q^{-22}+220q^{-24}+165q^{-26}-445q^{-28}+484q^{-30}-265q^{-32}-118q^{-34}+506q^{-36}-702q^{-38}+623q^{-40}-277q^{-42}-216q^{-44}+661q^{-46}-905q^{-48}+841q^{-50}-511q^{-52}+19q^{-54}+454q^{-56}-748q^{-58}+762q^{-60}-503q^{-62}+81q^{-64}+313q^{-66}-530q^{-68}+465q^{-70}-168q^{-72}-207q^{-74}+498q^{-76}-548q^{-78}+346q^{-80}+29q^{-82}-415q^{-84}+644q^{-86}-631q^{-88}+398q^{-90}-50q^{-92}-270q^{-94}+454q^{-96}-460q^{-98}+334q^{-100}-139q^{-102}-42q^{-104}+147q^{-106}-183q^{-108}+147q^{-110}-84q^{-112}+31q^{-114}+10q^{-116}-25q^{-118}+26q^{-120}-20q^{-122}+10q^{-124}-4q^{-126}+q^{-128}$

.

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["10 113"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-11t^{2}+26t-33+26t^{-1}-11t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 111, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{8}+5q^{7}-10q^{6}+14q^{5}-18q^{4}+19q^{3}-17q^{2}+14q-8+4q^{-1}-q^{-2}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$3z^{9}a^{-3}+3z^{9}a^{-5}+7z^{8}a^{-2}+16z^{8}a^{-4}+9z^{8}a^{-6}+7z^{7}a^{-1}+12z^{7}a^{-3}+15z^{7}a^{-5}+10z^{7}a^{-7}-5z^{6}a^{-2}-23z^{6}a^{-4}-9z^{6}a^{-6}+5z^{6}a^{-8}+4z^{6}+az^{5}-10z^{5}a^{-1}-30z^{5}a^{-3}-36z^{5}a^{-5}-16z^{5}a^{-7}+z^{5}a^{-9}-6z^{4}a^{-2}+z^{4}a^{-4}-4z^{4}a^{-6}-5z^{4}a^{-8}-6z^{4}-az^{3}+5z^{3}a^{-1}+17z^{3}a^{-3}+16z^{3}a^{-5}+5z^{3}a^{-7}+8z^{2}a^{-2}+8z^{2}a^{-4}+3z^{2}a^{-6}+3z^{2}-za^{-1}-za^{-3}+za^{-5}+za^{-7}-3a^{-2}-3a^{-4}-a^{-6}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a107, K11a347, ...}

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

Vassiliev invariants

V2 and V3:

(0, -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 $0$ $-8$ $0$ $-32$ $-8$ $0$ $-{\frac {176}{3}}$ $-{\frac {32}{3}}$ $-8$ $0$ $32$ $0$ $0$ $0$ $-{\frac {328}{3}}$ ${\frac {488}{3}}$ $-{\frac {112}{3}}$ $32$

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

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

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{23}-5q^{22}+5q^{21}+16q^{20}-41q^{19}+8q^{18}+83q^{17}-108q^{16}-27q^{15}+196q^{14}-159q^{13}-102q^{12}+295q^{11}-161q^{10}-174q^{9}+325q^{8}-116q^{7}-203q^{6}+269q^{5}-47q^{4}-171q^{3}+157q^{2}+4q-94+56q^{-1}+12q^{-2}-29q^{-3}+11q^{-4}+3q^{-5}-4q^{-6}+q^{-7}$
3	$-q^{45}+5q^{44}-5q^{43}-11q^{42}+11q^{41}+36q^{40}-14q^{39}-108q^{38}+17q^{37}+213q^{36}+49q^{35}-383q^{34}-197q^{33}+572q^{32}+462q^{31}-730q^{30}-844q^{29}+808q^{28}+1301q^{27}-767q^{26}-1784q^{25}+624q^{24}+2202q^{23}-364q^{22}-2554q^{21}+73q^{20}+2767q^{19}+255q^{18}-2867q^{17}-568q^{16}+2834q^{15}+853q^{14}-2660q^{13}-1113q^{12}+2385q^{11}+1283q^{10}-1976q^{9}-1388q^{8}+1525q^{7}+1347q^{6}-1024q^{5}-1230q^{4}+617q^{3}+974q^{2}-268q-715+76q^{-1}+449q^{-2}+23q^{-3}-252q^{-4}-39q^{-5}+118q^{-6}+33q^{-7}-54q^{-8}-13q^{-9}+20q^{-10}+4q^{-11}-6q^{-12}-3q^{-13}+4q^{-14}-q^{-15}$
4	$q^{74}-5q^{73}+5q^{72}+11q^{71}-16q^{70}-6q^{69}-30q^{68}+54q^{67}+103q^{66}-82q^{65}-112q^{64}-251q^{63}+212q^{62}+645q^{61}+48q^{60}-469q^{59}-1402q^{58}-26q^{57}+2075q^{56}+1563q^{55}-206q^{54}-4248q^{53}-2542q^{52}+3229q^{51}+5529q^{50}+3239q^{49}-7254q^{48}-8631q^{47}+897q^{46}+10069q^{45}+11259q^{44}-6767q^{43}-15986q^{42}-6381q^{41}+11329q^{40}+21118q^{39}-1369q^{38}-20470q^{37}-15829q^{36}+7975q^{35}+28515q^{34}+6390q^{33}-20505q^{32}-23517q^{31}+2084q^{30}+31572q^{29}+13360q^{28}-17291q^{27}-27794q^{26}-4111q^{25}+30657q^{24}+18367q^{23}-11987q^{22}-28602q^{21}-9929q^{20}+26018q^{19}+21092q^{18}-4872q^{17}-25517q^{16}-14666q^{15}+17731q^{14}+20332q^{13}+2697q^{12}-18207q^{11}-16185q^{10}+7772q^{9}+15153q^{8}+7485q^{7}-8838q^{6}-12951q^{5}+302q^{4}+7617q^{3}+7305q^{2}-1746q-6996-2153q^{-1}+1895q^{-2}+4049q^{-3}+829q^{-4}-2326q^{-5}-1348q^{-6}-201q^{-7}+1302q^{-8}+662q^{-9}-441q^{-10}-330q^{-11}-268q^{-12}+248q^{-13}+179q^{-14}-69q^{-15}-17q^{-16}-70q^{-17}+37q^{-18}+26q^{-19}-19q^{-20}+5q^{-21}-9q^{-22}+6q^{-23}+3q^{-24}-4q^{-25}+q^{-26}$
5	$-q^{110}+5q^{109}-5q^{108}-11q^{107}+16q^{106}+11q^{105}-10q^{103}-49q^{102}-53q^{101}+77q^{100}+193q^{99}+105q^{98}-147q^{97}-470q^{96}-463q^{95}+146q^{94}+1142q^{93}+1425q^{92}+120q^{91}-2076q^{90}-3380q^{89}-1743q^{88}+2924q^{87}+7090q^{86}+5594q^{85}-2529q^{84}-11815q^{83}-13351q^{82}-1683q^{81}+16722q^{80}+25488q^{79}+11986q^{78}-18233q^{77}-40936q^{76}-31019q^{75}+12436q^{74}+56673q^{73}+58646q^{72}+4708q^{71}-67378q^{70}-92461q^{69}-35372q^{68}+67642q^{67}+127438q^{66}+78290q^{65}-53411q^{64}-156999q^{63}-129444q^{62}+23740q^{61}+175703q^{60}+182272q^{59}+18982q^{58}-180144q^{57}-230663q^{56}-69435q^{55}+170524q^{54}+269064q^{53}+122008q^{52}-149290q^{51}-295846q^{50}-170921q^{49}+120958q^{48}+310323q^{47}+213088q^{46}-89127q^{45}-315143q^{44}-247013q^{43}+57310q^{42}+312043q^{41}+273152q^{40}-26342q^{39}-303206q^{38}-292904q^{37}-3679q^{36}+289392q^{35}+307134q^{34}+33780q^{33}-269756q^{32}-316379q^{31}-65076q^{30}+243562q^{29}+319348q^{28}+97142q^{27}-208866q^{26}-314188q^{25}-128881q^{24}+166215q^{23}+298125q^{22}+156560q^{21}-116345q^{20}-269958q^{19}-176317q^{18}+64083q^{17}+229332q^{16}+183631q^{15}-13836q^{14}-179900q^{13}-176795q^{12}-26835q^{11}+126211q^{10}+155697q^{9}+54716q^{8}-75683q^{7}-124947q^{6}-66442q^{5}+33970q^{4}+89488q^{3}+64870q^{2}-5068q-56439-53407q^{-1}-10729q^{-2}+29891q^{-3}+38155q^{-4}+15804q^{-5}-12198q^{-6}-23416q^{-7}-14367q^{-8}+2521q^{-9}+12382q^{-10}+10079q^{-11}+1302q^{-12}-5328q^{-13}-5903q^{-14}-2040q^{-15}+1877q^{-16}+2885q^{-17}+1431q^{-18}-405q^{-19}-1165q^{-20}-799q^{-21}-6q^{-22}+435q^{-23}+328q^{-24}+26q^{-25}-111q^{-26}-98q^{-27}-46q^{-28}+40q^{-29}+47q^{-30}-15q^{-31}-9q^{-32}+6q^{-33}-6q^{-34}+9q^{-36}-6q^{-37}-3q^{-38}+4q^{-39}-q^{-40}$
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, 113]]
Out[2]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[14, 6, 15, 5], X[20, 16, 1, 15], X[12, 7, 13, 8], X[8, 18, 9, 17], X[6, 19, 7, 20], X[16, 12, 17, 11], X[18, 13, 19, 14], X[2, 10, 3, 9]]
In[3]:=	GaussCode[Knot[10, 113]]
Out[3]=	GaussCode[1, -10, 2, -1, 3, -7, 5, -6, 10, -2, 8, -5, 9, -3, 4, -8, 6, -9, 7, -4]
In[4]:=	DTCode[Knot[10, 113]]
Out[4]=	DTCode[4, 10, 14, 12, 2, 16, 18, 20, 8, 6]
In[5]:=	br = BR[Knot[10, 113]]
Out[5]=	BR[4, {1, 1, 1, 2, -3, 2, -1, 2, -3, 2, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 113]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 113]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[10, 113]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 113]][t]
Out[10]=	2 11 26 2 3 -33 + -- - -- + -- + 26 t - 11 t + 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 113]][z]
Out[11]=	4 6 1 + z + 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 113], Knot[11, Alternating, 107], Knot[11, Alternating, 347]}
In[13]:=	{KnotDet[Knot[10, 113]], KnotSignature[Knot[10, 113]]}
Out[13]=	{111, 2}
In[14]:=	Jones[Knot[10, 113]][q]
Out[14]=	-2 4 2 3 4 5 6 7 8 -8 - q + - + 14 q - 17 q + 19 q - 18 q + 14 q - 10 q + 5 q - q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 113]}
In[16]:=	A2Invariant[Knot[10, 113]][q]
Out[16]=	-6 2 -2 2 4 6 10 12 14 -1 - q + -- - q + 5 q - 2 q + 4 q - 2 q + q - 5 q + 4 q 16 18 20 22 24 3 q - q - q + 3 q - q
In[17]:=	HOMFLYPT[Knot[10, 113]][a, z]
Out[17]=	2 2 4 4 4 6 6 -6 3 3 2 2 z 3 z 4 z z 2 z z z a - -- + -- - z - ---- + ---- - z - -- + -- + ---- + -- + -- 4 2 4 2 6 4 2 4 2 a a a a a a a a a
In[18]:=	Kauffman[Knot[10, 113]][a, z]
Out[18]=	2 2 2 3 -6 3 3 z z z z 2 3 z 8 z 8 z 5 z -a - -- - -- + -- + -- - -- - - + 3 z + ---- + ---- + ---- + ---- + 4 2 7 5 3 a 6 4 2 7 a a a a a a a a a 3 3 3 4 4 4 4 5 16 z 17 z 5 z 3 4 5 z 4 z z 6 z z ----- + ----- + ---- - a z - 6 z - ---- - ---- + -- - ---- + -- - 5 3 a 8 6 4 2 9 a a a a a a a 5 5 5 5 6 6 6 16 z 36 z 30 z 10 z 5 6 5 z 9 z 23 z ----- - ----- - ----- - ----- + a z + 4 z + ---- - ---- - ----- - 7 5 3 a 8 6 4 a a a a a a 6 7 7 7 7 8 8 8 9 5 z 10 z 15 z 12 z 7 z 9 z 16 z 7 z 3 z ---- + ----- + ----- + ----- + ---- + ---- + ----- + ---- + ---- + 2 7 5 3 a 6 4 2 5 a a a a a a a a 9 3 z ---- 3 a
In[19]:=	{Vassiliev[2][Knot[10, 113]], Vassiliev[3][Knot[10, 113]]}
Out[19]=	{0, -1}
In[20]:=	Kh[Knot[10, 113]][q, t]
Out[20]=	3 1 3 1 5 3 q 3 5 9 q + 6 q + ----- + ----- + ---- + --- + --- + 9 q t + 8 q t + 5 3 3 2 2 q t t q t q t q t 5 2 7 2 7 3 9 3 9 4 11 4 10 q t + 9 q t + 8 q t + 10 q t + 6 q t + 8 q t + 11 5 13 5 13 6 15 6 17 7 4 q t + 6 q t + q t + 4 q t + q t
In[21]:=	ColouredJones[Knot[10, 113], 2][q]
Out[21]=	-7 4 3 11 29 12 56 2 3 -94 + q - -- + -- + -- - -- + -- + -- + 4 q + 157 q - 171 q - 6 5 4 3 2 q q q q q q 4 5 6 7 8 9 10 47 q + 269 q - 203 q - 116 q + 325 q - 174 q - 161 q + 11 12 13 14 15 16 17 295 q - 102 q - 159 q + 196 q - 27 q - 108 q + 83 q + 18 19 20 21 22 23 8 q - 41 q + 16 q + 5 q - 5 q + q

See/edit the Rolfsen_Splice_Template. Back to the top.

10_112 10_114