K11a303

K11a302

K11a304

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a303 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X10,3,11,4 X18,5,19,6 X22,8,1,7 X16,10,17,9 X4,11,5,12 X8,14,9,13 X20,16,21,15 X12,18,13,17 X2,19,3,20 X14,22,15,21
Gauss code	1, -10, 2, -6, 3, -1, 4, -7, 5, -2, 6, -9, 7, -11, 8, -5, 9, -3, 10, -8, 11, -4
Dowker-Thistlethwaite code	6 10 18 22 16 4 8 20 12 2 14

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a303/ThurstonBennequinNumber Hyperbolic Volume 17.779 A-Polynomial See Data:K11a303/A-polynomial

[edit Notes for K11a303's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a303's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-3t^{3}+15t^{2}-34t+45-34t^{-1}+15t^{-2}-3t^{-3}$
Conway polynomial	$-3z^{6}-3z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 149, 0 }
Jones polynomial	$-q^{7}+4q^{6}-9q^{5}+15q^{4}-20q^{3}+24q^{2}-24q+21-16q^{-1}+10q^{-2}-4q^{-3}+q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$-2z^{6}a^{-2}-z^{6}+a^{2}z^{4}-6z^{4}a^{-2}+3z^{4}a^{-4}-z^{4}+a^{2}z^{2}-7z^{2}a^{-2}+6z^{2}a^{-4}-z^{2}a^{-6}+a^{2}-2a^{-2}+3a^{-4}-a^{-6}$
Kauffman polynomial (db, data sources)	$3z^{10}a^{-2}+3z^{10}a^{-4}+11z^{9}a^{-1}+17z^{9}a^{-3}+6z^{9}a^{-5}+20z^{8}a^{-2}+7z^{8}a^{-4}+4z^{8}a^{-6}+17z^{8}+16az^{7}-8z^{7}a^{-1}-41z^{7}a^{-3}-16z^{7}a^{-5}+z^{7}a^{-7}+10a^{2}z^{6}-69z^{6}a^{-2}-45z^{6}a^{-4}-13z^{6}a^{-6}-27z^{6}+4a^{3}z^{5}-22az^{5}-19z^{5}a^{-1}+18z^{5}a^{-3}+8z^{5}a^{-5}-3z^{5}a^{-7}+a^{4}z^{4}-8a^{2}z^{4}+58z^{4}a^{-2}+52z^{4}a^{-4}+14z^{4}a^{-6}+11z^{4}+11az^{3}+14z^{3}a^{-1}+5z^{3}a^{-3}+5z^{3}a^{-5}+3z^{3}a^{-7}+4a^{2}z^{2}-19z^{2}a^{-2}-20z^{2}a^{-4}-6z^{2}a^{-6}-z^{2}-2az-3za^{-1}-2za^{-3}-2za^{-5}-za^{-7}-a^{2}+2a^{-2}+3a^{-4}+a^{-6}$
The A2 invariant	Data:K11a303/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a303/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a303 Quantum Invariants.

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["K11a303"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-3t^{3}+15t^{2}-34t+45-34t^{-1}+15t^{-2}-3t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 149, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{7}+4q^{6}-9q^{5}+15q^{4}-20q^{3}+24q^{2}-24q+21-16q^{-1}+10q^{-2}-4q^{-3}+q^{-4}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-2z^{6}a^{-2}-z^{6}+a^{2}z^{4}-6z^{4}a^{-2}+3z^{4}a^{-4}-z^{4}+a^{2}z^{2}-7z^{2}a^{-2}+6z^{2}a^{-4}-z^{2}a^{-6}+a^{2}-2a^{-2}+3a^{-4}-a^{-6}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$3z^{10}a^{-2}+3z^{10}a^{-4}+11z^{9}a^{-1}+17z^{9}a^{-3}+6z^{9}a^{-5}+20z^{8}a^{-2}+7z^{8}a^{-4}+4z^{8}a^{-6}+17z^{8}+16az^{7}-8z^{7}a^{-1}-41z^{7}a^{-3}-16z^{7}a^{-5}+z^{7}a^{-7}+10a^{2}z^{6}-69z^{6}a^{-2}-45z^{6}a^{-4}-13z^{6}a^{-6}-27z^{6}+4a^{3}z^{5}-22az^{5}-19z^{5}a^{-1}+18z^{5}a^{-3}+8z^{5}a^{-5}-3z^{5}a^{-7}+a^{4}z^{4}-8a^{2}z^{4}+58z^{4}a^{-2}+52z^{4}a^{-4}+14z^{4}a^{-6}+11z^{4}+11az^{3}+14z^{3}a^{-1}+5z^{3}a^{-3}+5z^{3}a^{-5}+3z^{3}a^{-7}+4a^{2}z^{2}-19z^{2}a^{-2}-20z^{2}a^{-4}-6z^{2}a^{-6}-z^{2}-2az-3za^{-1}-2za^{-3}-2za^{-5}-za^{-7}-a^{2}+2a^{-2}+3a^{-4}+a^{-6}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11a303"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

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

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

Out[4]=

{ $-3t^{3}+15t^{2}-34t+45-34t^{-1}+15t^{-2}-3t^{-3}$, $-q^{7}+4q^{6}-9q^{5}+15q^{4}-20q^{3}+24q^{2}-24q+21-16q^{-1}+10q^{-2}-4q^{-3}+q^{-4}$ }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{}

Vassiliev invariants

V2 and V3:

(-1, 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 $-4$ $8$ $8$ ${\frac {178}{3}}$ ${\frac {86}{3}}$ $-32$ ${\frac {80}{3}}$ $-{\frac {64}{3}}$ $40$ $-{\frac {32}{3}}$ $32$ $-{\frac {712}{3}}$ $-{\frac {344}{3}}$ $-{\frac {4831}{30}}$ ${\frac {22}{15}}$ $-{\frac {12902}{45}}$ ${\frac {2815}{18}}$ $-{\frac {1951}{30}}$

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=$0 is the signature of K11a303. 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 7 χ 15                       1 -1 13                     3   3 11                   6 1   -5 9                 9 3     6 7               11 6       -5 5             13 9         4 3           11 11           0 1         10 13             -3 -1       7 12               5 -3     3 9                 -6 -5   1 7                   6 -7   3                     -3 -9 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=0$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=3$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

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

Modifying This Page

Read me first: Modifying Knot Pages. See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate). See/edit the Hoste-Thistlethwaite_Splice_Base (expert). Back to the top.

K11a302 K11a304