K11a34

K11a33

K11a35

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a34 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X14,5,15,6 X2837 X16,9,17,10 X18,12,19,11 X20,14,21,13 X6,15,7,16 X22,17,1,18 X12,20,13,19 X10,22,11,21
Gauss code	1, -4, 2, -1, 3, -8, 4, -2, 5, -11, 6, -10, 7, -3, 8, -5, 9, -6, 10, -7, 11, -9
Dowker-Thistlethwaite code	4 8 14 2 16 18 20 6 22 12 10

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a34/ThurstonBennequinNumber Hyperbolic Volume 14.9742 A-Polynomial See Data:K11a34/A-polynomial

[edit Notes for K11a34's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a34's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+5t^{3}-14t^{2}+25t-29+25t^{-1}-14t^{-2}+5t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-3z^{6}-4z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 119, 2 }
Jones polynomial	$-q^{8}+4q^{7}-8q^{6}+13q^{5}-17q^{4}+19q^{3}-19q^{2}+16q-11+7q^{-1}-3q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$-z^{8}a^{-2}-6z^{6}a^{-2}+2z^{6}a^{-4}+z^{6}-15z^{4}a^{-2}+8z^{4}a^{-4}-z^{4}a^{-6}+4z^{4}-17z^{2}a^{-2}+11z^{2}a^{-4}-2z^{2}a^{-6}+6z^{2}-7a^{-2}+5a^{-4}-a^{-6}+4$
Kauffman polynomial (db, data sources)	$z^{10}a^{-2}+z^{10}a^{-4}+4z^{9}a^{-1}+8z^{9}a^{-3}+4z^{9}a^{-5}+12z^{8}a^{-2}+14z^{8}a^{-4}+7z^{8}a^{-6}+5z^{8}+3az^{7}-5z^{7}a^{-1}-9z^{7}a^{-3}+6z^{7}a^{-5}+7z^{7}a^{-7}+a^{2}z^{6}-41z^{6}a^{-2}-37z^{6}a^{-4}-7z^{6}a^{-6}+4z^{6}a^{-8}-14z^{6}-8az^{5}-4z^{5}a^{-1}-9z^{5}a^{-3}-25z^{5}a^{-5}-11z^{5}a^{-7}+z^{5}a^{-9}-3a^{2}z^{4}+53z^{4}a^{-2}+39z^{4}a^{-4}-2z^{4}a^{-6}-6z^{4}a^{-8}+15z^{4}+5az^{3}+7z^{3}a^{-1}+19z^{3}a^{-3}+23z^{3}a^{-5}+5z^{3}a^{-7}-z^{3}a^{-9}+2a^{2}z^{2}-32z^{2}a^{-2}-20z^{2}a^{-4}+z^{2}a^{-6}+2z^{2}a^{-8}-11z^{2}-az-3za^{-1}-7za^{-3}-7za^{-5}-2za^{-7}+7a^{-2}+5a^{-4}+a^{-6}+4$
The A2 invariant	$q^{8}-q^{6}+3q^{4}+3q^{-2}-5q^{-4}+2q^{-6}-3q^{-8}+2q^{-12}-2q^{-14}+4q^{-16}-q^{-18}+q^{-22}-q^{-24}$
The G2 invariant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{46}-2 q^{44}+5 q^{42}-9 q^{40}+11 q^{38}-12 q^{36}+5 q^{34}+12 q^{32}-34 q^{30}+62 q^{28}-83 q^{26}+79 q^{24}-44 q^{22}-30 q^{20}+134 q^{18}-222 q^{16}+269 q^{14}-226 q^{12}+92 q^{10}+111 q^8-317 q^6+451 q^4-442 q^2+284-18 q^{-2} -260 q^{-4} +446 q^{-6} -463 q^{-8} +316 q^{-10} -56 q^{-12} -204 q^{-14} +343 q^{-16} -319 q^{-18} +124 q^{-20} +141 q^{-22} -360 q^{-24} +430 q^{-26} -308 q^{-28} +18 q^{-30} +321 q^{-32} -592 q^{-34} +670 q^{-36} -522 q^{-38} +179 q^{-40} +230 q^{-42} -564 q^{-44} +705 q^{-46} -601 q^{-48} +312 q^{-50} +52 q^{-52} -347 q^{-54} +467 q^{-56} -377 q^{-58} +147 q^{-60} +125 q^{-62} -297 q^{-64} +311 q^{-66} -157 q^{-68} -85 q^{-70} +313 q^{-72} -425 q^{-74} +382 q^{-76} -198 q^{-78} -62 q^{-80} +292 q^{-82} -425 q^{-84} +423 q^{-86} -298 q^{-88} +108 q^{-90} +81 q^{-92} -221 q^{-94} +271 q^{-96} -242 q^{-98} +159 q^{-100} -55 q^{-102} -32 q^{-104} +82 q^{-106} -98 q^{-108} +82 q^{-110} -49 q^{-112} +21 q^{-114} +3 q^{-116} -14 q^{-118} +15 q^{-120} -13 q^{-122} +7 q^{-124} -3 q^{-126} + q^{-128} }

Further Quantum Invariants[show]

K11a34 Quantum Invariants.

Computer Talk[show]

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["K11a34"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+5t^{3}-14t^{2}+25t-29+25t^{-1}-14t^{-2}+5t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 119, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{8}+4q^{7}-8q^{6}+13q^{5}-17q^{4}+19q^{3}-19q^{2}+16q-11+7q^{-1}-3q^{-2}+q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{8}a^{-2}-6z^{6}a^{-2}+2z^{6}a^{-4}+z^{6}-15z^{4}a^{-2}+8z^{4}a^{-4}-z^{4}a^{-6}+4z^{4}-17z^{2}a^{-2}+11z^{2}a^{-4}-2z^{2}a^{-6}+6z^{2}-7a^{-2}+5a^{-4}-a^{-6}+4$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-2}+z^{10}a^{-4}+4z^{9}a^{-1}+8z^{9}a^{-3}+4z^{9}a^{-5}+12z^{8}a^{-2}+14z^{8}a^{-4}+7z^{8}a^{-6}+5z^{8}+3az^{7}-5z^{7}a^{-1}-9z^{7}a^{-3}+6z^{7}a^{-5}+7z^{7}a^{-7}+a^{2}z^{6}-41z^{6}a^{-2}-37z^{6}a^{-4}-7z^{6}a^{-6}+4z^{6}a^{-8}-14z^{6}-8az^{5}-4z^{5}a^{-1}-9z^{5}a^{-3}-25z^{5}a^{-5}-11z^{5}a^{-7}+z^{5}a^{-9}-3a^{2}z^{4}+53z^{4}a^{-2}+39z^{4}a^{-4}-2z^{4}a^{-6}-6z^{4}a^{-8}+15z^{4}+5az^{3}+7z^{3}a^{-1}+19z^{3}a^{-3}+23z^{3}a^{-5}+5z^{3}a^{-7}-z^{3}a^{-9}+2a^{2}z^{2}-32z^{2}a^{-2}-20z^{2}a^{-4}+z^{2}a^{-6}+2z^{2}a^{-8}-11z^{2}-az-3za^{-1}-7za^{-3}-7za^{-5}-2za^{-7}+7a^{-2}+5a^{-4}+a^{-6}+4$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a158,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a89,}

Computer Talk[show]

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["K11a34"];

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]=

{ $-t^{4}+5t^{3}-14t^{2}+25t-29+25t^{-1}-14t^{-2}+5t^{-3}-t^{-4}$, $-q^{8}+4q^{7}-8q^{6}+13q^{5}-17q^{4}+19q^{3}-19q^{2}+16q-11+7q^{-1}-3q^{-2}+q^{-3}$ }

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]=

{K11a158,}

In[6]:=

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

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

Out[6]=

{K11a89,}

Vassiliev invariants

V2 and V3:

(-2, -1)

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 $-8$ $-8$ $32$ ${\frac {260}{3}}$ ${\frac {148}{3}}$ $64$ ${\frac {496}{3}}$ ${\frac {160}{3}}$ $24$ $-{\frac {256}{3}}$ $32$ $-{\frac {2080}{3}}$ $-{\frac {1184}{3}}$ $-{\frac {10111}{15}}$ ${\frac {2588}{5}}$ $-{\frac {54484}{45}}$ ${\frac {1471}{9}}$ $-{\frac {4591}{15}}$

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=$2 is the signature of K11a34. 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                   5 1   -4 11                 8 3     5 9               9 5       -4 7             10 8         2 5           9 9           0 3         7 10             -3 1       5 10               5 -1     2 6                 -4 -3   1 5                   4 -5   2                     -2 -7 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a33 K11a35