K11n9

K11n8

K11n10

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n9 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8394 X10,6,11,5 X14,8,15,7 X2,9,3,10 X11,19,12,18 X6,14,7,13 X15,21,16,20 X17,1,18,22 X19,13,20,12 X21,17,22,16
Gauss code	1, -5, 2, -1, 3, -7, 4, -2, 5, -3, -6, 10, 7, -4, -8, 11, -9, 6, -10, 8, -11, 9
Dowker-Thistlethwaite code	4 8 10 14 2 -18 6 -20 -22 -12 -16

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 3 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n9/ThurstonBennequinNumber Hyperbolic Volume 10.3175 A-Polynomial See Data:K11n9/A-polynomial

[edit Notes for K11n9's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $4$ Rasmussen s-Invariant -6

[edit Notes for K11n9's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+3t^{3}-t^{2}-4t+7-4t^{-1}-t^{-2}+3t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-5z^{6}-3z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 5, 4 }
Jones polynomial	$-q^{11}+2q^{10}-2q^{9}+2q^{8}-2q^{7}+q^{6}-q^{4}+2q^{3}-q^{2}+q$
HOMFLY-PT polynomial (db, data sources)	$-z^{8}a^{-6}+z^{6}a^{-4}-7z^{6}a^{-6}+z^{6}a^{-8}+6z^{4}a^{-4}-16z^{4}a^{-6}+7z^{4}a^{-8}+10z^{2}a^{-4}-17z^{2}a^{-6}+12z^{2}a^{-8}-2z^{2}a^{-10}+5a^{-4}-8a^{-6}+6a^{-8}-2a^{-10}$
Kauffman polynomial (db, data sources)	$z^{9}a^{-5}+z^{9}a^{-7}+z^{8}a^{-4}+3z^{8}a^{-6}+2z^{8}a^{-8}-6z^{7}a^{-5}-6z^{7}a^{-7}+z^{7}a^{-9}+z^{7}a^{-11}-7z^{6}a^{-4}-21z^{6}a^{-6}-15z^{6}a^{-8}+z^{6}a^{-10}+2z^{6}a^{-12}+9z^{5}a^{-5}+6z^{5}a^{-7}-7z^{5}a^{-9}-3z^{5}a^{-11}+z^{5}a^{-13}+16z^{4}a^{-4}+43z^{4}a^{-6}+31z^{4}a^{-8}-3z^{4}a^{-10}-7z^{4}a^{-12}-2z^{3}a^{-5}+5z^{3}a^{-7}+11z^{3}a^{-9}+z^{3}a^{-11}-3z^{3}a^{-13}-15z^{2}a^{-4}-32z^{2}a^{-6}-22z^{2}a^{-8}-z^{2}a^{-10}+4z^{2}a^{-12}-2za^{-5}-4za^{-7}-4za^{-9}-za^{-11}+za^{-13}+5a^{-4}+8a^{-6}+6a^{-8}+2a^{-10}$
The A2 invariant	$q^{-4}+q^{-6}+q^{-8}+2q^{-10}+q^{-12}-q^{-14}-q^{-16}-q^{-18}-2q^{-20}+q^{-22}+2q^{-26}+q^{-32}-2q^{-34}$
The G2 invariant	Data:K11n9/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n9 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["K11n9"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+3t^{3}-t^{2}-4t+7-4t^{-1}-t^{-2}+3t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{8}-5z^{6}-3z^{4}+3z^{2}+1$

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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 5, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{8}a^{-6}+z^{6}a^{-4}-7z^{6}a^{-6}+z^{6}a^{-8}+6z^{4}a^{-4}-16z^{4}a^{-6}+7z^{4}a^{-8}+10z^{2}a^{-4}-17z^{2}a^{-6}+12z^{2}a^{-8}-2z^{2}a^{-10}+5a^{-4}-8a^{-6}+6a^{-8}-2a^{-10}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-5}+z^{9}a^{-7}+z^{8}a^{-4}+3z^{8}a^{-6}+2z^{8}a^{-8}-6z^{7}a^{-5}-6z^{7}a^{-7}+z^{7}a^{-9}+z^{7}a^{-11}-7z^{6}a^{-4}-21z^{6}a^{-6}-15z^{6}a^{-8}+z^{6}a^{-10}+2z^{6}a^{-12}+9z^{5}a^{-5}+6z^{5}a^{-7}-7z^{5}a^{-9}-3z^{5}a^{-11}+z^{5}a^{-13}+16z^{4}a^{-4}+43z^{4}a^{-6}+31z^{4}a^{-8}-3z^{4}a^{-10}-7z^{4}a^{-12}-2z^{3}a^{-5}+5z^{3}a^{-7}+11z^{3}a^{-9}+z^{3}a^{-11}-3z^{3}a^{-13}-15z^{2}a^{-4}-32z^{2}a^{-6}-22z^{2}a^{-8}-z^{2}a^{-10}+4z^{2}a^{-12}-2za^{-5}-4za^{-7}-4za^{-9}-za^{-11}+za^{-13}+5a^{-4}+8a^{-6}+6a^{-8}+2a^{-10}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

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["K11n9"];

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]=

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

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]=

{}

Vassiliev invariants

V2 and V3:

(3, 7)

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 $12$ $56$ $72$ $318$ $58$ $672$ ${\frac {5264}{3}}$ ${\frac {992}{3}}$ $248$ $288$ $1568$ $3816$ $696$ ${\frac {97551}{10}}$ ${\frac {5054}{15}}$ ${\frac {57542}{15}}$ ${\frac {593}{6}}$ ${\frac {4751}{10}}$

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 K11n9. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 23                       1 -1 21                     1   1 19                   1 1   0 17               1 2 1     0 15             1 2 1       0 13           1 2 2         -1 11         1 2 2           1 9       1 1 1             -1 7     1 1 1               1 5   1 2                   1 3                         0 1 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11n8 K11n10