K11a1

10_165

K11a2

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a1 at Knotilus!

K11a1.

A graph, Knot K11a1.

Knot presentations

Planar diagram presentation	X4251 X8394 X10,6,11,5 X14,7,15,8 X2,9,3,10 X16,12,17,11 X20,14,21,13 X6,15,7,16 X22,18,1,17 X12,20,13,19 X18,22,19,21
Gauss code	1, -5, 2, -1, 3, -8, 4, -2, 5, -3, 6, -10, 7, -4, 8, -6, 9, -11, 10, -7, 11, -9
Dowker-Thistlethwaite code	4 8 10 14 2 16 20 6 22 12 18

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:K11a1/ThurstonBennequinNumber Hyperbolic Volume 15.6439 A-Polynomial See Data:K11a1/A-polynomial

[edit Notes for K11a1's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant -2

[edit Notes for K11a1's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-12t^{2}+30t-39+30t^{-1}-12t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 127, 2 }
Jones polynomial	$-q^{8}+4q^{7}-9q^{6}+14q^{5}-18q^{4}+21q^{3}-20q^{2}+17q-12+7q^{-1}-3q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+z^{6}a^{-4}+z^{4}a^{-2}+2z^{4}a^{-4}-z^{4}a^{-6}-2z^{4}+a^{2}z^{2}+3z^{2}a^{-4}-z^{2}a^{-6}-3z^{2}+a^{2}+2a^{-4}-a^{-6}-1$
Kauffman polynomial (db, data sources)	$z^{10}a^{-2}+z^{10}a^{-4}+3z^{9}a^{-1}+8z^{9}a^{-3}+5z^{9}a^{-5}+11z^{8}a^{-2}+16z^{8}a^{-4}+9z^{8}a^{-6}+4z^{8}+3az^{7}+3z^{7}a^{-1}-4z^{7}a^{-3}+4z^{7}a^{-5}+8z^{7}a^{-7}+a^{2}z^{6}-25z^{6}a^{-2}-36z^{6}a^{-4}-14z^{6}a^{-6}+4z^{6}a^{-8}-6z^{6}-8az^{5}-18z^{5}a^{-1}-16z^{5}a^{-3}-20z^{5}a^{-5}-13z^{5}a^{-7}+z^{5}a^{-9}-3a^{2}z^{4}+15z^{4}a^{-2}+28z^{4}a^{-4}+9z^{4}a^{-6}-5z^{4}a^{-8}-2z^{4}+7az^{3}+15z^{3}a^{-1}+16z^{3}a^{-3}+16z^{3}a^{-5}+7z^{3}a^{-7}-z^{3}a^{-9}+3a^{2}z^{2}-3z^{2}a^{-2}-10z^{2}a^{-4}-4z^{2}a^{-6}+z^{2}a^{-8}+5z^{2}-2az-4za^{-1}-4za^{-3}-4za^{-5}-2za^{-7}-a^{2}+2a^{-4}+a^{-6}-1$
The A2 invariant	Data:K11a1/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a1/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a1 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["K11a1"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-12t^{2}+30t-39+30t^{-1}-12t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$2z^{6}+1$

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]=

{ 127, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{8}+4q^{7}-9q^{6}+14q^{5}-18q^{4}+21q^{3}-20q^{2}+17q-12+7q^{-1}-3q^{-2}+q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-2}+z^{10}a^{-4}+3z^{9}a^{-1}+8z^{9}a^{-3}+5z^{9}a^{-5}+11z^{8}a^{-2}+16z^{8}a^{-4}+9z^{8}a^{-6}+4z^{8}+3az^{7}+3z^{7}a^{-1}-4z^{7}a^{-3}+4z^{7}a^{-5}+8z^{7}a^{-7}+a^{2}z^{6}-25z^{6}a^{-2}-36z^{6}a^{-4}-14z^{6}a^{-6}+4z^{6}a^{-8}-6z^{6}-8az^{5}-18z^{5}a^{-1}-16z^{5}a^{-3}-20z^{5}a^{-5}-13z^{5}a^{-7}+z^{5}a^{-9}-3a^{2}z^{4}+15z^{4}a^{-2}+28z^{4}a^{-4}+9z^{4}a^{-6}-5z^{4}a^{-8}-2z^{4}+7az^{3}+15z^{3}a^{-1}+16z^{3}a^{-3}+16z^{3}a^{-5}+7z^{3}a^{-7}-z^{3}a^{-9}+3a^{2}z^{2}-3z^{2}a^{-2}-10z^{2}a^{-4}-4z^{2}a^{-6}+z^{2}a^{-8}+5z^{2}-2az-4za^{-1}-4za^{-3}-4za^{-5}-2za^{-7}-a^{2}+2a^{-4}+a^{-6}-1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a122, K11a149,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a149,}

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["K11a1"];

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]=

{ $2t^{3}-12t^{2}+30t-39+30t^{-1}-12t^{-2}+2t^{-3}$, $-q^{8}+4q^{7}-9q^{6}+14q^{5}-18q^{4}+21q^{3}-20q^{2}+17q-12+7q^{-1}-3q^{-2}+q^{-3}$ }

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]=

{K11a122, K11a149,}

In[6]:=

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

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

Out[6]=

{K11a149,}

Vassiliev invariants

V2 and V3:

(0, 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 $0$ $16$ $0$ $32$ $0$ $0$ ${\frac {64}{3}}$ $-{\frac {128}{3}}$ $16$ $0$ $128$ $0$ $0$ $208$ $-{\frac {544}{3}}$ $224$ $-16$ $48$

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 K11a1. 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 17                       1 -1 15                     3   3 13                   6 1   -5 11                 8 3     5 9               10 6       -4 7             11 8         3 5           9 10           1 3         8 11             -3 1       5 10               5 -1     2 7                 -5 -3   1 5                   4 -5   2                     -2 -7 1                       1

Integral Khovanov Homology (db, data source)

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

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.

10_165 K11a2