K11a9

K11a8

K11a10

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a9 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a9/ThurstonBennequinNumber Hyperbolic Volume 11.385 A-Polynomial See Data:K11a9/A-polynomial

[edit Notes for K11a9's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a9's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-5t^{3}+10t^{2}-11t+11-11t^{-1}+10t^{-2}-5t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+3z^{6}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 65, 4 }
Jones polynomial	$-q^{9}+3q^{8}-5q^{7}+7q^{6}-9q^{5}+10q^{4}-9q^{3}+8q^{2}-6q+4-2q^{-1}+q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}a^{-4}-2z^{6}a^{-2}+6z^{6}a^{-4}-z^{6}a^{-6}-10z^{4}a^{-2}+13z^{4}a^{-4}-4z^{4}a^{-6}+z^{4}-14z^{2}a^{-2}+14z^{2}a^{-4}-4z^{2}a^{-6}+4z^{2}-6a^{-2}+6a^{-4}-2a^{-6}+3$
Kauffman polynomial (db, data sources)	$z^{10}a^{-2}+z^{10}a^{-4}+2z^{9}a^{-1}+6z^{9}a^{-3}+4z^{9}a^{-5}+5z^{8}a^{-4}+6z^{8}a^{-6}+z^{8}-11z^{7}a^{-1}-28z^{7}a^{-3}-11z^{7}a^{-5}+6z^{7}a^{-7}-20z^{6}a^{-2}-36z^{6}a^{-4}-16z^{6}a^{-6}+6z^{6}a^{-8}-6z^{6}+19z^{5}a^{-1}+38z^{5}a^{-3}+5z^{5}a^{-5}-9z^{5}a^{-7}+5z^{5}a^{-9}+44z^{4}a^{-2}+54z^{4}a^{-4}+12z^{4}a^{-6}-7z^{4}a^{-8}+3z^{4}a^{-10}+12z^{4}-12z^{3}a^{-1}-17z^{3}a^{-3}-z^{3}a^{-5}-z^{3}a^{-7}-4z^{3}a^{-9}+z^{3}a^{-11}-30z^{2}a^{-2}-29z^{2}a^{-4}-7z^{2}a^{-6}+z^{2}a^{-8}-z^{2}a^{-10}-10z^{2}+2za^{-1}+3za^{-3}+za^{-5}+za^{-7}+za^{-9}+6a^{-2}+6a^{-4}+2a^{-6}+3$
The A2 invariant	$q^{6}+q^{4}+1-q^{-2}-q^{-6}-q^{-8}+2q^{-10}-q^{-12}+3q^{-14}-q^{-22}+q^{-24}-q^{-26}$
The G2 invariant	$q^{26}-q^{24}+4q^{22}-5q^{20}+6q^{18}-5q^{16}+q^{14}+10q^{12}-19q^{10}+30q^{8}-29q^{6}+19q^{4}+4q^{2}-31+54q^{-2}-58q^{-4}+43q^{-6}-13q^{-8}-24q^{-10}+50q^{-12}-58q^{-14}+42q^{-16}-14q^{-18}-18q^{-20}+34q^{-22}-35q^{-24}+12q^{-26}+13q^{-28}-30q^{-30}+36q^{-32}-26q^{-34}+q^{-36}+26q^{-38}-50q^{-40}+57q^{-42}-47q^{-44}+21q^{-46}+16q^{-48}-43q^{-50}+60q^{-52}-55q^{-54}+38q^{-56}-5q^{-58}-24q^{-60}+39q^{-62}-35q^{-64}+19q^{-66}+5q^{-68}-15q^{-70}+19q^{-72}-7q^{-74}-8q^{-76}+18q^{-78}-21q^{-80}+16q^{-82}-6q^{-84}-9q^{-86}+17q^{-88}-19q^{-90}+18q^{-92}-15q^{-94}+9q^{-96}-5q^{-98}-5q^{-100}+12q^{-102}-22q^{-104}+25q^{-106}-20q^{-108}+14q^{-110}-q^{-112}-13q^{-114}+21q^{-116}-25q^{-118}+21q^{-120}-12q^{-122}+2q^{-124}+6q^{-126}-12q^{-128}+14q^{-130}-11q^{-132}+8q^{-134}-2q^{-136}-q^{-138}+2q^{-140}-4q^{-142}+3q^{-144}-2q^{-146}+q^{-148}$

Further Quantum Invariants

K11a9 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["K11a9"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-5t^{3}+10t^{2}-11t+11-11t^{-1}+10t^{-2}-5t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{8}+3z^{6}+1$

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]=

{ 65, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{9}+3q^{8}-5q^{7}+7q^{6}-9q^{5}+10q^{4}-9q^{3}+8q^{2}-6q+4-2q^{-1}+q^{-2}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}a^{-4}-2z^{6}a^{-2}+6z^{6}a^{-4}-z^{6}a^{-6}-10z^{4}a^{-2}+13z^{4}a^{-4}-4z^{4}a^{-6}+z^{4}-14z^{2}a^{-2}+14z^{2}a^{-4}-4z^{2}a^{-6}+4z^{2}-6a^{-2}+6a^{-4}-2a^{-6}+3$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-2}+z^{10}a^{-4}+2z^{9}a^{-1}+6z^{9}a^{-3}+4z^{9}a^{-5}+5z^{8}a^{-4}+6z^{8}a^{-6}+z^{8}-11z^{7}a^{-1}-28z^{7}a^{-3}-11z^{7}a^{-5}+6z^{7}a^{-7}-20z^{6}a^{-2}-36z^{6}a^{-4}-16z^{6}a^{-6}+6z^{6}a^{-8}-6z^{6}+19z^{5}a^{-1}+38z^{5}a^{-3}+5z^{5}a^{-5}-9z^{5}a^{-7}+5z^{5}a^{-9}+44z^{4}a^{-2}+54z^{4}a^{-4}+12z^{4}a^{-6}-7z^{4}a^{-8}+3z^{4}a^{-10}+12z^{4}-12z^{3}a^{-1}-17z^{3}a^{-3}-z^{3}a^{-5}-z^{3}a^{-7}-4z^{3}a^{-9}+z^{3}a^{-11}-30z^{2}a^{-2}-29z^{2}a^{-4}-7z^{2}a^{-6}+z^{2}a^{-8}-z^{2}a^{-10}-10z^{2}+2za^{-1}+3za^{-3}+za^{-5}+za^{-7}+za^{-9}+6a^{-2}+6a^{-4}+2a^{-6}+3$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a140,}

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["K11a9"];

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]=

{ $t^{4}-5t^{3}+10t^{2}-11t+11-11t^{-1}+10t^{-2}-5t^{-3}+t^{-4}$, $-q^{9}+3q^{8}-5q^{7}+7q^{6}-9q^{5}+10q^{4}-9q^{3}+8q^{2}-6q+4-2q^{-1}+q^{-2}$ }

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]=

{K11a140,}

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 $0$ $16$ $0$ $48$ $0$ $0$ ${\frac {64}{3}}$ $-{\frac {224}{3}}$ $16$ $0$ $128$ $0$ $0$ $88$ $-352$ $240$ $-24$ $72$

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=$4 is the signature of K11a9. 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 19                       1 -1 17                     2   2 15                   3 1   -2 13                 4 2     2 11               5 3       -2 9             5 4         1 7           4 5           1 5         4 5             -1 3       3 5               2 1     1 3                 -2 -1   1 3                   2 -3   1                     -1 -5 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a8 K11a10