K11a223

K11a222

K11a224

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a223 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X12,4,13,3 X18,6,19,5 X20,8,21,7 X16,10,17,9 X2,12,3,11 X22,13,1,14 X8,16,9,15 X10,18,11,17 X6,20,7,19 X14,21,15,22
Gauss code	1, -6, 2, -1, 3, -10, 4, -8, 5, -9, 6, -2, 7, -11, 8, -5, 9, -3, 10, -4, 11, -7
Dowker-Thistlethwaite code	4 12 18 20 16 2 22 8 10 6 14

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a223/ThurstonBennequinNumber Hyperbolic Volume 12.6981 A-Polynomial See Data:K11a223/A-polynomial

[edit Notes for K11a223'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 K11a223's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+5t^{3}-10t^{2}+14t-15+14t^{-1}-10t^{-2}+5t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-3z^{6}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 75, 6 }
Jones polynomial	$-q^{12}+3q^{11}-5q^{10}+8q^{9}-11q^{8}+11q^{7}-11q^{6}+10q^{5}-7q^{4}+5q^{3}-2q^{2}+q$
HOMFLY-PT polynomial (db, data sources)	$-z^{8}a^{-6}+z^{6}a^{-4}-6z^{6}a^{-6}+2z^{6}a^{-8}+5z^{4}a^{-4}-13z^{4}a^{-6}+9z^{4}a^{-8}-z^{4}a^{-10}+8z^{2}a^{-4}-13z^{2}a^{-6}+11z^{2}a^{-8}-3z^{2}a^{-10}+4a^{-4}-5a^{-6}+3a^{-8}-a^{-10}$
Kauffman polynomial (db, data sources)	$z^{10}a^{-6}+z^{10}a^{-8}+2z^{9}a^{-5}+6z^{9}a^{-7}+4z^{9}a^{-9}+z^{8}a^{-4}+z^{8}a^{-6}+8z^{8}a^{-8}+8z^{8}a^{-10}-10z^{7}a^{-5}-22z^{7}a^{-7}-3z^{7}a^{-9}+9z^{7}a^{-11}-6z^{6}a^{-4}-22z^{6}a^{-6}-44z^{6}a^{-8}-21z^{6}a^{-10}+7z^{6}a^{-12}+15z^{5}a^{-5}+15z^{5}a^{-7}-25z^{5}a^{-9}-20z^{5}a^{-11}+5z^{5}a^{-13}+13z^{4}a^{-4}+43z^{4}a^{-6}+55z^{4}a^{-8}+14z^{4}a^{-10}-8z^{4}a^{-12}+3z^{4}a^{-14}-5z^{3}a^{-5}+9z^{3}a^{-7}+30z^{3}a^{-9}+12z^{3}a^{-11}-3z^{3}a^{-13}+z^{3}a^{-15}-12z^{2}a^{-4}-27z^{2}a^{-6}-22z^{2}a^{-8}-6z^{2}a^{-10}-z^{2}a^{-14}-3za^{-5}-6za^{-7}-6za^{-9}-3za^{-11}+4a^{-4}+5a^{-6}+3a^{-8}+a^{-10}$
The A2 invariant	$q^{-4}+2q^{-8}+q^{-10}+2q^{-14}-2q^{-16}+2q^{-18}-2q^{-20}-q^{-22}-2q^{-26}+2q^{-28}+q^{-32}-q^{-36}$
The G2 invariant	$q^{-22}-q^{-24}+5q^{-26}-7q^{-28}+10q^{-30}-9q^{-32}+3q^{-34}+15q^{-36}-31q^{-38}+47q^{-40}-46q^{-42}+27q^{-44}+13q^{-46}-57q^{-48}+91q^{-50}-90q^{-52}+63q^{-54}-4q^{-56}-56q^{-58}+93q^{-60}-96q^{-62}+63q^{-64}-7q^{-66}-48q^{-68}+69q^{-70}-58q^{-72}+19q^{-74}+30q^{-76}-66q^{-78}+71q^{-80}-48q^{-82}-4q^{-84}+57q^{-86}-105q^{-88}+112q^{-90}-80q^{-92}+26q^{-94}+44q^{-96}-100q^{-98}+121q^{-100}-100q^{-102}+47q^{-104}+16q^{-106}-71q^{-108}+85q^{-110}-57q^{-112}+14q^{-114}+31q^{-116}-52q^{-118}+47q^{-120}-14q^{-122}-29q^{-124}+54q^{-126}-61q^{-128}+47q^{-130}-10q^{-132}-25q^{-134}+48q^{-136}-53q^{-138}+47q^{-140}-31q^{-142}+8q^{-144}+9q^{-146}-27q^{-148}+33q^{-150}-32q^{-152}+26q^{-154}-13q^{-156}+4q^{-158}+7q^{-160}-19q^{-162}+21q^{-164}-21q^{-166}+15q^{-168}-7q^{-170}+q^{-172}+6q^{-174}-11q^{-176}+13q^{-178}-10q^{-180}+7q^{-182}-2q^{-184}-q^{-186}+2q^{-188}-4q^{-190}+3q^{-192}-2q^{-194}+q^{-196}$

Further Quantum Invariants

K11a223 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["K11a223"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+5t^{3}-10t^{2}+14t-15+14t^{-1}-10t^{-2}+5t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 75, 6 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{12}+3q^{11}-5q^{10}+8q^{9}-11q^{8}+11q^{7}-11q^{6}+10q^{5}-7q^{4}+5q^{3}-2q^{2}+q$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{8}a^{-6}+z^{6}a^{-4}-6z^{6}a^{-6}+2z^{6}a^{-8}+5z^{4}a^{-4}-13z^{4}a^{-6}+9z^{4}a^{-8}-z^{4}a^{-10}+8z^{2}a^{-4}-13z^{2}a^{-6}+11z^{2}a^{-8}-3z^{2}a^{-10}+4a^{-4}-5a^{-6}+3a^{-8}-a^{-10}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-6}+z^{10}a^{-8}+2z^{9}a^{-5}+6z^{9}a^{-7}+4z^{9}a^{-9}+z^{8}a^{-4}+z^{8}a^{-6}+8z^{8}a^{-8}+8z^{8}a^{-10}-10z^{7}a^{-5}-22z^{7}a^{-7}-3z^{7}a^{-9}+9z^{7}a^{-11}-6z^{6}a^{-4}-22z^{6}a^{-6}-44z^{6}a^{-8}-21z^{6}a^{-10}+7z^{6}a^{-12}+15z^{5}a^{-5}+15z^{5}a^{-7}-25z^{5}a^{-9}-20z^{5}a^{-11}+5z^{5}a^{-13}+13z^{4}a^{-4}+43z^{4}a^{-6}+55z^{4}a^{-8}+14z^{4}a^{-10}-8z^{4}a^{-12}+3z^{4}a^{-14}-5z^{3}a^{-5}+9z^{3}a^{-7}+30z^{3}a^{-9}+12z^{3}a^{-11}-3z^{3}a^{-13}+z^{3}a^{-15}-12z^{2}a^{-4}-27z^{2}a^{-6}-22z^{2}a^{-8}-6z^{2}a^{-10}-z^{2}a^{-14}-3za^{-5}-6za^{-7}-6za^{-9}-3za^{-11}+4a^{-4}+5a^{-6}+3a^{-8}+a^{-10}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n148,}

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["K11a223"];

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]=

{ $-t^{4}+5t^{3}-10t^{2}+14t-15+14t^{-1}-10t^{-2}+5t^{-3}-t^{-4}$, $-q^{12}+3q^{11}-5q^{10}+8q^{9}-11q^{8}+11q^{7}-11q^{6}+10q^{5}-7q^{4}+5q^{3}-2q^{2}+q$ }

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]=

{K11n148,}

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, 6)

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 $12$ $48$ $72$ $238$ $34$ $576$ $1248$ $128$ $240$ $288$ $1152$ $2856$ $408$ ${\frac {66591}{10}}$ $-{\frac {10826}{15}}$ ${\frac {48302}{15}}$ ${\frac {1505}{6}}$ ${\frac {4191}{10}}$

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=$6 is the signature of K11a223. 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 25                       1 -1 23                     2   2 21                   3 1   -2 19                 5 2     3 17               6 3       -3 15             5 5         0 13           6 6           0 11         4 5             -1 9       3 6               3 7     2 4                 -2 5   1 4                   3 3   1                     -1 1 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a222 K11a224