K11a285

K11a284

K11a286

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a285 at Knotilus!

Knot presentations

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

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:K11a285/ThurstonBennequinNumber Hyperbolic Volume 18.3328 A-Polynomial See Data:K11a285/A-polynomial

[edit Notes for K11a285'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 K11a285's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+14t^{2}-38t+53-38t^{-1}+14t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}+2z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 161, 0 }
Jones polynomial	$-q^{5}+4q^{4}-9q^{3}+16q^{2}-22q+26-26q^{-1}+23q^{-2}-17q^{-3}+11q^{-4}-5q^{-5}+q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{6}-z^{6}+a^{4}z^{4}-a^{2}z^{4}+2z^{4}a^{-2}-a^{2}z^{2}+z^{2}a^{-2}-z^{2}a^{-4}+z^{2}+1$
Kauffman polynomial (db, data sources)	$4a^{2}z^{10}+4z^{10}+10a^{3}z^{9}+20az^{9}+10z^{9}a^{-1}+10a^{4}z^{8}+10a^{2}z^{8}+11z^{8}a^{-2}+11z^{8}+5a^{5}z^{7}-19a^{3}z^{7}-47az^{7}-15z^{7}a^{-1}+8z^{7}a^{-3}+a^{6}z^{6}-22a^{4}z^{6}-40a^{2}z^{6}-18z^{6}a^{-2}+4z^{6}a^{-4}-39z^{6}-9a^{5}z^{5}+9a^{3}z^{5}+41az^{5}+11z^{5}a^{-1}-11z^{5}a^{-3}+z^{5}a^{-5}-a^{6}z^{4}+11a^{4}z^{4}+32a^{2}z^{4}+13z^{4}a^{-2}-5z^{4}a^{-4}+38z^{4}+2a^{5}z^{3}-6a^{3}z^{3}-17az^{3}-4z^{3}a^{-1}+4z^{3}a^{-3}-z^{3}a^{-5}-a^{4}z^{2}-8a^{2}z^{2}-4z^{2}a^{-2}+z^{2}a^{-4}-12z^{2}+a^{5}z+3a^{3}z+3az+za^{-1}+1$
The A2 invariant	Data:K11a285/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a285/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a285 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["K11a285"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{3}+14t^{2}-38t+53-38t^{-1}+14t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 161, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{5}+4q^{4}-9q^{3}+16q^{2}-22q+26-26q^{-1}+23q^{-2}-17q^{-3}+11q^{-4}-5q^{-5}+q^{-6}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$4a^{2}z^{10}+4z^{10}+10a^{3}z^{9}+20az^{9}+10z^{9}a^{-1}+10a^{4}z^{8}+10a^{2}z^{8}+11z^{8}a^{-2}+11z^{8}+5a^{5}z^{7}-19a^{3}z^{7}-47az^{7}-15z^{7}a^{-1}+8z^{7}a^{-3}+a^{6}z^{6}-22a^{4}z^{6}-40a^{2}z^{6}-18z^{6}a^{-2}+4z^{6}a^{-4}-39z^{6}-9a^{5}z^{5}+9a^{3}z^{5}+41az^{5}+11z^{5}a^{-1}-11z^{5}a^{-3}+z^{5}a^{-5}-a^{6}z^{4}+11a^{4}z^{4}+32a^{2}z^{4}+13z^{4}a^{-2}-5z^{4}a^{-4}+38z^{4}+2a^{5}z^{3}-6a^{3}z^{3}-17az^{3}-4z^{3}a^{-1}+4z^{3}a^{-3}-z^{3}a^{-5}-a^{4}z^{2}-8a^{2}z^{2}-4z^{2}a^{-2}+z^{2}a^{-4}-12z^{2}+a^{5}z+3a^{3}z+3az+za^{-1}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a138,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a138,}

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["K11a285"];

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]=

{ $-2t^{3}+14t^{2}-38t+53-38t^{-1}+14t^{-2}-2t^{-3}$, $-q^{5}+4q^{4}-9q^{3}+16q^{2}-22q+26-26q^{-1}+23q^{-2}-17q^{-3}+11q^{-4}-5q^{-5}+q^{-6}$ }

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]=

{K11a138,}

In[6]:=

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

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

Out[6]=

{K11a138,}

Vassiliev invariants

V2 and V3:

(0, 0)

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 $0$ $0$ $0$ $-16$ $-16$ $0$ $32$ $-64$ $96$ $0$ $0$ $0$ $0$ $-40$ ${\frac {784}{3}}$ $-{\frac {800}{3}}$ $-{\frac {104}{3}}$ $-88$

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 K11a285. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     3   3 7                   6 1   -5 5                 10 3     7 3               12 6       -6 1             14 10         4 -1           13 13           0 -3         10 13             -3 -5       7 13               6 -7     4 10                 -6 -9   1 7                   6 -11   4                     -4 -13 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-2$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-1$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=0$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{14}$ $r=1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a284 K11a286