K11a33

K11a32

K11a34

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a33 at Knotilus!

Knot presentations

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a33/ThurstonBennequinNumber Hyperbolic Volume 13.8946 A-Polynomial See Data:K11a33/A-polynomial

[edit Notes for K11a33'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 K11a33's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+5t^{3}-12t^{2}+19t-21+19t^{-1}-12t^{-2}+5t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-3z^{6}-2z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 95, -2 }
Jones polynomial	$-q^{4}+3q^{3}-6q^{2}+10q-12+15q^{-1}-15q^{-2}+13q^{-3}-10q^{-4}+6q^{-5}-3q^{-6}+q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{8}+a^{4}z^{6}-6a^{2}z^{6}+2z^{6}+4a^{4}z^{4}-14a^{2}z^{4}-z^{4}a^{-2}+9z^{4}+5a^{4}z^{2}-15a^{2}z^{2}-3z^{2}a^{-2}+13z^{2}+2a^{4}-6a^{2}-2a^{-2}+7$
Kauffman polynomial (db, data sources)	$a^{2}z^{10}+z^{10}+4a^{3}z^{9}+7az^{9}+3z^{9}a^{-1}+6a^{4}z^{8}+10a^{2}z^{8}+3z^{8}a^{-2}+7z^{8}+6a^{5}z^{7}-2a^{3}z^{7}-16az^{7}-7z^{7}a^{-1}+z^{7}a^{-3}+5a^{6}z^{6}-8a^{4}z^{6}-37a^{2}z^{6}-12z^{6}a^{-2}-36z^{6}+3a^{7}z^{5}-6a^{5}z^{5}-9a^{3}z^{5}-4z^{5}a^{-1}-4z^{5}a^{-3}+a^{8}z^{4}-5a^{6}z^{4}+5a^{4}z^{4}+42a^{2}z^{4}+15z^{4}a^{-2}+46z^{4}-3a^{7}z^{3}+2a^{5}z^{3}+10a^{3}z^{3}+13az^{3}+13z^{3}a^{-1}+5z^{3}a^{-3}-a^{8}z^{2}+2a^{6}z^{2}-4a^{4}z^{2}-24a^{2}z^{2}-8z^{2}a^{-2}-25z^{2}+a^{7}z-4a^{3}z-6az-5za^{-1}-2za^{-3}+2a^{4}+6a^{2}+2a^{-2}+7$
The A2 invariant	$q^{20}-q^{18}+2q^{16}-q^{14}-q^{12}+q^{10}-4q^{8}+2q^{6}-2q^{4}+2q^{2}+3+3q^{-4}-q^{-6}-q^{-12}$
The G2 invariant	$q^{114}-2q^{112}+4q^{110}-6q^{108}+5q^{106}-4q^{104}-2q^{102}+11q^{100}-20q^{98}+28q^{96}-30q^{94}+21q^{92}-4q^{90}-18q^{88}+46q^{86}-64q^{84}+71q^{82}-62q^{80}+34q^{78}+5q^{76}-50q^{74}+94q^{72}-120q^{70}+122q^{68}-91q^{66}+28q^{64}+51q^{62}-120q^{60}+164q^{58}-153q^{56}+91q^{54}+2q^{52}-95q^{50}+142q^{48}-123q^{46}+48q^{44}+53q^{42}-131q^{40}+140q^{38}-78q^{36}-43q^{34}+164q^{32}-242q^{30}+220q^{28}-117q^{26}-45q^{24}+200q^{22}-292q^{20}+287q^{18}-190q^{16}+33q^{14}+123q^{12}-226q^{10}+246q^{8}-169q^{6}+45q^{4}+87q^{2}-158+157q^{-2}-71q^{-4}-46q^{-6}+152q^{-8}-189q^{-10}+143q^{-12}-26q^{-14}-108q^{-16}+214q^{-18}-235q^{-20}+177q^{-22}-63q^{-24}-68q^{-26}+158q^{-28}-189q^{-30}+156q^{-32}-82q^{-34}+q^{-36}+56q^{-38}-82q^{-40}+74q^{-42}-48q^{-44}+19q^{-46}+3q^{-48}-15q^{-50}+14q^{-52}-11q^{-54}+6q^{-56}-2q^{-58}+q^{-60}$

Further Quantum Invariants

K11a33 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["K11a33"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+5t^{3}-12t^{2}+19t-21+19t^{-1}-12t^{-2}+5t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 95, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{4}+3q^{3}-6q^{2}+10q-12+15q^{-1}-15q^{-2}+13q^{-3}-10q^{-4}+6q^{-5}-3q^{-6}+q^{-7}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{2}z^{8}+a^{4}z^{6}-6a^{2}z^{6}+2z^{6}+4a^{4}z^{4}-14a^{2}z^{4}-z^{4}a^{-2}+9z^{4}+5a^{4}z^{2}-15a^{2}z^{2}-3z^{2}a^{-2}+13z^{2}+2a^{4}-6a^{2}-2a^{-2}+7$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{2}z^{10}+z^{10}+4a^{3}z^{9}+7az^{9}+3z^{9}a^{-1}+6a^{4}z^{8}+10a^{2}z^{8}+3z^{8}a^{-2}+7z^{8}+6a^{5}z^{7}-2a^{3}z^{7}-16az^{7}-7z^{7}a^{-1}+z^{7}a^{-3}+5a^{6}z^{6}-8a^{4}z^{6}-37a^{2}z^{6}-12z^{6}a^{-2}-36z^{6}+3a^{7}z^{5}-6a^{5}z^{5}-9a^{3}z^{5}-4z^{5}a^{-1}-4z^{5}a^{-3}+a^{8}z^{4}-5a^{6}z^{4}+5a^{4}z^{4}+42a^{2}z^{4}+15z^{4}a^{-2}+46z^{4}-3a^{7}z^{3}+2a^{5}z^{3}+10a^{3}z^{3}+13az^{3}+13z^{3}a^{-1}+5z^{3}a^{-3}-a^{8}z^{2}+2a^{6}z^{2}-4a^{4}z^{2}-24a^{2}z^{2}-8z^{2}a^{-2}-25z^{2}+a^{7}z-4a^{3}z-6az-5za^{-1}-2za^{-3}+2a^{4}+6a^{2}+2a^{-2}+7$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_116, K11a7, K11a82,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a82,}

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["K11a33"];

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]=

{ $-t^{4}+5t^{3}-12t^{2}+19t-21+19t^{-1}-12t^{-2}+5t^{-3}-t^{-4}$, $-q^{4}+3q^{3}-6q^{2}+10q-12+15q^{-1}-15q^{-2}+13q^{-3}-10q^{-4}+6q^{-5}-3q^{-6}+q^{-7}$ }

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]=

{10_116, K11a7, K11a82,}

In[6]:=

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

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

Out[6]=

{K11a82,}

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 $0$ $16$ $0$ $0$ $16$ $0$ $-{\frac {320}{3}}$ $-{\frac {128}{3}}$ $-80$ $0$ $128$ $0$ $0$ $448$ ${\frac {64}{3}}$ ${\frac {656}{3}}$ ${\frac {272}{3}}$ $-16$

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 K11a33. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11a32 K11a34