K11a203

K11a202

K11a204

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a203 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 4 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a203/ThurstonBennequinNumber Hyperbolic Volume 10.2613 A-Polynomial See Data:K11a203/A-polynomial

[edit Notes for K11a203'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 K11a203's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+5t^{3}-9t^{2}+11t-11+11t^{-1}-9t^{-2}+5t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-3z^{6}+z^{4}+4z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 63, 6 }
Jones polynomial	$-q^{12}+3q^{11}-5q^{10}+7q^{9}-9q^{8}+9q^{7}-9q^{6}+8q^{5}-5q^{4}+4q^{3}-2q^{2}+q$
HOMFLY-PT polynomial (db, data sources)	$-z^{8}a^{-6}+z^{6}a^{-4}-6z^{6}a^{-6}+2z^{6}a^{-8}+5z^{4}a^{-4}-12z^{4}a^{-6}+9z^{4}a^{-8}-z^{4}a^{-10}+7z^{2}a^{-4}-10z^{2}a^{-6}+10z^{2}a^{-8}-3z^{2}a^{-10}+3a^{-4}-3a^{-6}+2a^{-8}-a^{-10}$
Kauffman polynomial (db, data sources)	$z^{10}a^{-6}+z^{10}a^{-8}+2z^{9}a^{-5}+5z^{9}a^{-7}+3z^{9}a^{-9}+z^{8}a^{-4}-z^{8}a^{-6}+3z^{8}a^{-8}+5z^{8}a^{-10}-11z^{7}a^{-5}-22z^{7}a^{-7}-5z^{7}a^{-9}+6z^{7}a^{-11}-6z^{6}a^{-4}-14z^{6}a^{-6}-25z^{6}a^{-8}-11z^{6}a^{-10}+6z^{6}a^{-12}+18z^{5}a^{-5}+26z^{5}a^{-7}-7z^{5}a^{-9}-10z^{5}a^{-11}+5z^{5}a^{-13}+12z^{4}a^{-4}+32z^{4}a^{-6}+35z^{4}a^{-8}+5z^{4}a^{-10}-7z^{4}a^{-12}+3z^{4}a^{-14}-8z^{3}a^{-5}-7z^{3}a^{-7}+8z^{3}a^{-9}+2z^{3}a^{-11}-4z^{3}a^{-13}+z^{3}a^{-15}-10z^{2}a^{-4}-20z^{2}a^{-6}-16z^{2}a^{-8}-4z^{2}a^{-10}+z^{2}a^{-12}-z^{2}a^{-14}-za^{-5}+za^{-13}+3a^{-4}+3a^{-6}+2a^{-8}+a^{-10}$
The A2 invariant	$q^{-4}+q^{-8}+q^{-10}+2q^{-14}-q^{-16}+2q^{-18}-q^{-20}-q^{-22}-2q^{-26}+q^{-28}+q^{-32}-q^{-36}$
The G2 invariant	$q^{-22}-q^{-24}+4q^{-26}-5q^{-28}+6q^{-30}-4q^{-32}+q^{-34}+10q^{-36}-18q^{-38}+27q^{-40}-26q^{-42}+14q^{-44}+6q^{-46}-28q^{-48}+46q^{-50}-47q^{-52}+36q^{-54}-8q^{-56}-20q^{-58}+41q^{-60}-47q^{-62}+36q^{-64}-13q^{-66}-10q^{-68}+25q^{-70}-25q^{-72}+16q^{-74}+2q^{-76}-17q^{-78}+25q^{-80}-22q^{-82}+3q^{-84}+14q^{-86}-36q^{-88}+46q^{-90}-38q^{-92}+20q^{-94}+6q^{-96}-34q^{-98}+51q^{-100}-53q^{-102}+34q^{-104}-9q^{-106}-17q^{-108}+33q^{-110}-32q^{-112}+23q^{-114}-8q^{-116}-4q^{-118}+11q^{-120}-11q^{-122}+2q^{-124}+5q^{-126}-9q^{-128}+11q^{-130}-5q^{-132}-q^{-134}+7q^{-136}-13q^{-138}+18q^{-140}-19q^{-142}+15q^{-144}-7q^{-146}-6q^{-148}+17q^{-150}-25q^{-152}+26q^{-154}-21q^{-156}+12q^{-158}+q^{-160}-16q^{-162}+22q^{-164}-24q^{-166}+20q^{-168}-11q^{-170}+2q^{-172}+6q^{-174}-12q^{-176}+14q^{-178}-11q^{-180}+8q^{-182}-2q^{-184}-q^{-186}+2q^{-188}-4q^{-190}+3q^{-192}-2q^{-194}+q^{-196}$

Further Quantum Invariants

K11a203 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["K11a203"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+5t^{3}-9t^{2}+11t-11+11t^{-1}-9t^{-2}+5t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 63, 6 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-6}+z^{10}a^{-8}+2z^{9}a^{-5}+5z^{9}a^{-7}+3z^{9}a^{-9}+z^{8}a^{-4}-z^{8}a^{-6}+3z^{8}a^{-8}+5z^{8}a^{-10}-11z^{7}a^{-5}-22z^{7}a^{-7}-5z^{7}a^{-9}+6z^{7}a^{-11}-6z^{6}a^{-4}-14z^{6}a^{-6}-25z^{6}a^{-8}-11z^{6}a^{-10}+6z^{6}a^{-12}+18z^{5}a^{-5}+26z^{5}a^{-7}-7z^{5}a^{-9}-10z^{5}a^{-11}+5z^{5}a^{-13}+12z^{4}a^{-4}+32z^{4}a^{-6}+35z^{4}a^{-8}+5z^{4}a^{-10}-7z^{4}a^{-12}+3z^{4}a^{-14}-8z^{3}a^{-5}-7z^{3}a^{-7}+8z^{3}a^{-9}+2z^{3}a^{-11}-4z^{3}a^{-13}+z^{3}a^{-15}-10z^{2}a^{-4}-20z^{2}a^{-6}-16z^{2}a^{-8}-4z^{2}a^{-10}+z^{2}a^{-12}-z^{2}a^{-14}-za^{-5}+za^{-13}+3a^{-4}+3a^{-6}+2a^{-8}+a^{-10}$

"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["K11a203"];

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]=

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

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]=

{}

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:

(4, 9)

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 $16$ $72$ $128$ ${\frac {1160}{3}}$ ${\frac {160}{3}}$ $1152$ $2256$ $320$ $328$ ${\frac {2048}{3}}$ $2592$ ${\frac {18560}{3}}$ ${\frac {2560}{3}}$ ${\frac {203222}{15}}$ $-{\frac {1168}{15}}$ ${\frac {238448}{45}}$ ${\frac {2218}{9}}$ ${\frac {10022}{15}}$

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 K11a203. 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                 4 2     2 17               5 3       -2 15             4 4         0 13           5 5           0 11         3 4             -1 9       2 5               3 7     2 3                 -1 5   1 3                   2 3   1                     -1 1 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a202 K11a204