K11a255

K11a254

K11a256

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a255 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X8394 X12,6,13,5 X20,8,21,7 X16,9,17,10 X18,11,19,12 X22,13,1,14 X4,16,5,15 X10,17,11,18 X2,19,3,20 X14,21,15,22
Gauss code	1, -10, 2, -8, 3, -1, 4, -2, 5, -9, 6, -3, 7, -11, 8, -5, 9, -6, 10, -4, 11, -7
Dowker-Thistlethwaite code	6 8 12 20 16 18 22 4 10 2 14

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 1 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a255/ThurstonBennequinNumber Hyperbolic Volume 17.3074 A-Polynomial See Data:K11a255/A-polynomial

[edit Notes for K11a255's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a255's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+6t^{3}-17t^{2}+30t-35+30t^{-1}-17t^{-2}+6t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-2z^{6}-z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 143, -2 }
Jones polynomial	$q^{3}-4q^{2}+9q-14+20q^{-1}-23q^{-2}+23q^{-3}-20q^{-4}+15q^{-5}-9q^{-6}+4q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{8}+2a^{4}z^{6}-5a^{2}z^{6}+z^{6}-a^{6}z^{4}+7a^{4}z^{4}-10a^{2}z^{4}+3z^{4}-2a^{6}z^{2}+8a^{4}z^{2}-9a^{2}z^{2}+3z^{2}-a^{6}+3a^{4}-3a^{2}+2$
Kauffman polynomial (db, data sources)	$3a^{4}z^{10}+3a^{2}z^{10}+8a^{5}z^{9}+16a^{3}z^{9}+8az^{9}+10a^{6}z^{8}+11a^{4}z^{8}+9a^{2}z^{8}+8z^{8}+8a^{7}z^{7}-9a^{5}z^{7}-40a^{3}z^{7}-19az^{7}+4z^{7}a^{-1}+4a^{8}z^{6}-16a^{6}z^{6}-39a^{4}z^{6}-42a^{2}z^{6}+z^{6}a^{-2}-22z^{6}+a^{9}z^{5}-12a^{7}z^{5}+a^{5}z^{5}+38a^{3}z^{5}+15az^{5}-9z^{5}a^{-1}-5a^{8}z^{4}+10a^{6}z^{4}+44a^{4}z^{4}+50a^{2}z^{4}-2z^{4}a^{-2}+19z^{4}-a^{9}z^{3}+5a^{7}z^{3}+2a^{5}z^{3}-13a^{3}z^{3}-6az^{3}+3z^{3}a^{-1}+a^{8}z^{2}-5a^{6}z^{2}-20a^{4}z^{2}-22a^{2}z^{2}-8z^{2}-a^{7}z-a^{5}z+a^{3}z+az+a^{6}+3a^{4}+3a^{2}+2$
The A2 invariant	$-q^{24}+q^{22}-2q^{18}+4q^{16}-3q^{14}+2q^{12}+q^{10}-3q^{8}+4q^{6}-5q^{4}+4q^{2}-q^{-2}+3q^{-4}-2q^{-6}+q^{-8}$
The G2 invariant	Data:K11a255/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a255 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["K11a255"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+6t^{3}-17t^{2}+30t-35+30t^{-1}-17t^{-2}+6t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{8}-2z^{6}-z^{4}+1$

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]=

{ 143, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{3}-4q^{2}+9q-14+20q^{-1}-23q^{-2}+23q^{-3}-20q^{-4}+15q^{-5}-9q^{-6}+4q^{-7}-q^{-8}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{2}z^{8}+2a^{4}z^{6}-5a^{2}z^{6}+z^{6}-a^{6}z^{4}+7a^{4}z^{4}-10a^{2}z^{4}+3z^{4}-2a^{6}z^{2}+8a^{4}z^{2}-9a^{2}z^{2}+3z^{2}-a^{6}+3a^{4}-3a^{2}+2$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$3a^{4}z^{10}+3a^{2}z^{10}+8a^{5}z^{9}+16a^{3}z^{9}+8az^{9}+10a^{6}z^{8}+11a^{4}z^{8}+9a^{2}z^{8}+8z^{8}+8a^{7}z^{7}-9a^{5}z^{7}-40a^{3}z^{7}-19az^{7}+4z^{7}a^{-1}+4a^{8}z^{6}-16a^{6}z^{6}-39a^{4}z^{6}-42a^{2}z^{6}+z^{6}a^{-2}-22z^{6}+a^{9}z^{5}-12a^{7}z^{5}+a^{5}z^{5}+38a^{3}z^{5}+15az^{5}-9z^{5}a^{-1}-5a^{8}z^{4}+10a^{6}z^{4}+44a^{4}z^{4}+50a^{2}z^{4}-2z^{4}a^{-2}+19z^{4}-a^{9}z^{3}+5a^{7}z^{3}+2a^{5}z^{3}-13a^{3}z^{3}-6az^{3}+3z^{3}a^{-1}+a^{8}z^{2}-5a^{6}z^{2}-20a^{4}z^{2}-22a^{2}z^{2}-8z^{2}-a^{7}z-a^{5}z+a^{3}z+az+a^{6}+3a^{4}+3a^{2}+2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a79,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a79,}

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["K11a255"];

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]=

{ $-t^{4}+6t^{3}-17t^{2}+30t-35+30t^{-1}-17t^{-2}+6t^{-3}-t^{-4}$, $q^{3}-4q^{2}+9q-14+20q^{-1}-23q^{-2}+23q^{-3}-20q^{-4}+15q^{-5}-9q^{-6}+4q^{-7}-q^{-8}$ }

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]=

{K11a79,}

In[6]:=

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

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

Out[6]=

{K11a79,}

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 $0$ $-8$ $0$ $32$ $8$ $0$ $-{\frac {176}{3}}$ ${\frac {160}{3}}$ $-72$ $0$ $32$ $0$ $0$ $128$ $-{\frac {824}{3}}$ ${\frac {664}{3}}$ ${\frac {304}{3}}$ $32$

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 K11a255. 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 4 χ 7                       1 1 5                     3   -3 3                   6 1   5 1                 8 3     -5 -1               12 6       6 -3             12 9         -3 -5           11 11           0 -7         9 12             3 -9       6 11               -5 -11     3 9                 6 -13   1 6                   -5 -15   3                     3 -17 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-4$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-3$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=-1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=0$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{12}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a254 K11a256