K11a87

K11a86

K11a88

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a87 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X12,6,13,5 X16,8,17,7 X18,10,19,9 X2,11,3,12 X20,13,21,14 X8,16,9,15 X6,18,7,17 X22,19,1,20 X14,21,15,22
Gauss code	1, -6, 2, -1, 3, -9, 4, -8, 5, -2, 6, -3, 7, -11, 8, -4, 9, -5, 10, -7, 11, -10
Dowker-Thistlethwaite code	4 10 12 16 18 2 20 8 6 22 14

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a87/ThurstonBennequinNumber Hyperbolic Volume 15.1381 A-Polynomial See Data:K11a87/A-polynomial

[edit Notes for K11a87's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a87's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+11t^{2}-28t+39-28t^{-1}+11t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\left\{2,t^{2}+t+1\right\}$
Determinant and Signature	{ 121, 0 }
Jones polynomial	$q^{6}-3q^{5}+7q^{4}-12q^{3}+16q^{2}-19q+20-17q^{-1}+13q^{-2}-8q^{-3}+4q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-2}-z^{6}+2a^{2}z^{4}-3z^{4}a^{-2}+z^{4}a^{-4}-z^{4}-a^{4}z^{2}+2a^{2}z^{2}-6z^{2}a^{-2}+2z^{2}a^{-4}+z^{2}-4a^{-2}+2a^{-4}+3$
Kauffman polynomial (db, data sources)	$z^{10}a^{-2}+z^{10}+4az^{9}+8z^{9}a^{-1}+4z^{9}a^{-3}+7a^{2}z^{8}+12z^{8}a^{-2}+5z^{8}a^{-4}+14z^{8}+7a^{3}z^{7}+6az^{7}-8z^{7}a^{-1}-4z^{7}a^{-3}+3z^{7}a^{-5}+4a^{4}z^{6}-7a^{2}z^{6}-37z^{6}a^{-2}-13z^{6}a^{-4}+z^{6}a^{-6}-34z^{6}+a^{5}z^{5}-11a^{3}z^{5}-22az^{5}-8z^{5}a^{-1}-6z^{5}a^{-3}-8z^{5}a^{-5}-6a^{4}z^{4}-a^{2}z^{4}+41z^{4}a^{-2}+12z^{4}a^{-4}-3z^{4}a^{-6}+31z^{4}-a^{5}z^{3}+5a^{3}z^{3}+16az^{3}+12z^{3}a^{-1}+8z^{3}a^{-3}+6z^{3}a^{-5}+2a^{4}z^{2}+a^{2}z^{2}-21z^{2}a^{-2}-7z^{2}a^{-4}+2z^{2}a^{-6}-13z^{2}-a^{3}z-3az-3za^{-1}-3za^{-3}-2za^{-5}+4a^{-2}+2a^{-4}+3$
The A2 invariant	Data:K11a87/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a87/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a87 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["K11a87"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{3}+11t^{2}-28t+39-28t^{-1}+11t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

$\left\{2,t^{2}+t+1\right\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 121, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{6}-3q^{5}+7q^{4}-12q^{3}+16q^{2}-19q+20-17q^{-1}+13q^{-2}-8q^{-3}+4q^{-4}-q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-2}+z^{10}+4az^{9}+8z^{9}a^{-1}+4z^{9}a^{-3}+7a^{2}z^{8}+12z^{8}a^{-2}+5z^{8}a^{-4}+14z^{8}+7a^{3}z^{7}+6az^{7}-8z^{7}a^{-1}-4z^{7}a^{-3}+3z^{7}a^{-5}+4a^{4}z^{6}-7a^{2}z^{6}-37z^{6}a^{-2}-13z^{6}a^{-4}+z^{6}a^{-6}-34z^{6}+a^{5}z^{5}-11a^{3}z^{5}-22az^{5}-8z^{5}a^{-1}-6z^{5}a^{-3}-8z^{5}a^{-5}-6a^{4}z^{4}-a^{2}z^{4}+41z^{4}a^{-2}+12z^{4}a^{-4}-3z^{4}a^{-6}+31z^{4}-a^{5}z^{3}+5a^{3}z^{3}+16az^{3}+12z^{3}a^{-1}+8z^{3}a^{-3}+6z^{3}a^{-5}+2a^{4}z^{2}+a^{2}z^{2}-21z^{2}a^{-2}-7z^{2}a^{-4}+2z^{2}a^{-6}-13z^{2}-a^{3}z-3az-3za^{-1}-3za^{-3}-2za^{-5}+4a^{-2}+2a^{-4}+3$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a28, K11a96,}

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["K11a87"];

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]=

{ $-2t^{3}+11t^{2}-28t+39-28t^{-1}+11t^{-2}-2t^{-3}$, $q^{6}-3q^{5}+7q^{4}-12q^{3}+16q^{2}-19q+20-17q^{-1}+13q^{-2}-8q^{-3}+4q^{-4}-q^{-5}$ }

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]=

{K11a28, K11a96,}

Vassiliev invariants

V2 and V3:

(-2, -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 $-8$ $-16$ $32$ ${\frac {116}{3}}$ ${\frac {76}{3}}$ $128$ ${\frac {800}{3}}$ ${\frac {224}{3}}$ $80$ $-{\frac {256}{3}}$ $128$ $-{\frac {928}{3}}$ $-{\frac {608}{3}}$ ${\frac {1769}{15}}$ ${\frac {628}{5}}$ $-{\frac {10204}{45}}$ ${\frac {1111}{9}}$ $-{\frac {1591}{15}}$

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 K11a87. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a86 K11a88