K11a7

K11a6

K11a8

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a7 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8394 X10,6,11,5 X16,8,17,7 X2,9,3,10 X18,11,19,12 X20,13,21,14 X6,16,7,15 X22,17,1,18 X12,19,13,20 X14,21,15,22
Gauss code	1, -5, 2, -1, 3, -8, 4, -2, 5, -3, 6, -10, 7, -11, 8, -4, 9, -6, 10, -7, 11, -9
Dowker-Thistlethwaite code	4 8 10 16 2 18 20 6 22 12 14

A Braid Representative
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:K11a7/ThurstonBennequinNumber Hyperbolic Volume 13.5338 A-Polynomial See Data:K11a7/A-polynomial

[edit Notes for K11a7'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 K11a7'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	$q^{3}-3q^{2}+6q-9+13q^{-1}-15q^{-2}+15q^{-3}-13q^{-4}+10q^{-5}-6q^{-6}+3q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{8}+2a^{4}z^{6}-6a^{2}z^{6}+z^{6}-a^{6}z^{4}+9a^{4}z^{4}-14a^{2}z^{4}+4z^{4}-3a^{6}z^{2}+13a^{4}z^{2}-15a^{2}z^{2}+5z^{2}-2a^{6}+6a^{4}-6a^{2}+3$
Kauffman polynomial (db, data sources)	$a^{4}z^{10}+a^{2}z^{10}+3a^{5}z^{9}+7a^{3}z^{9}+4az^{9}+4a^{6}z^{8}+7a^{4}z^{8}+8a^{2}z^{8}+5z^{8}+4a^{7}z^{7}-2a^{5}z^{7}-19a^{3}z^{7}-10az^{7}+3z^{7}a^{-1}+3a^{8}z^{6}-4a^{6}z^{6}-26a^{4}z^{6}-37a^{2}z^{6}+z^{6}a^{-2}-17z^{6}+a^{9}z^{5}-5a^{7}z^{5}+a^{5}z^{5}+21a^{3}z^{5}+5az^{5}-9z^{5}a^{-1}-6a^{8}z^{4}+a^{6}z^{4}+41a^{4}z^{4}+55a^{2}z^{4}-3z^{4}a^{-2}+18z^{4}-2a^{9}z^{3}-a^{7}z^{3}-2a^{5}z^{3}-6a^{3}z^{3}+2az^{3}+5z^{3}a^{-1}+3a^{8}z^{2}-4a^{6}z^{2}-27a^{4}z^{2}-31a^{2}z^{2}+z^{2}a^{-2}-10z^{2}+a^{9}z+a^{7}z-az-za^{-1}+2a^{6}+6a^{4}+6a^{2}+3$
The A2 invariant	$-q^{24}-q^{18}+3q^{16}-q^{14}+2q^{12}+q^{10}-2q^{8}+2q^{6}-4q^{4}+2q^{2}+2q^{-4}-q^{-6}+q^{-8}$
The G2 invariant	$q^{128}-2q^{126}+5q^{124}-8q^{122}+8q^{120}-6q^{118}-2q^{116}+16q^{114}-29q^{112}+41q^{110}-42q^{108}+27q^{106}-2q^{104}-37q^{102}+75q^{100}-101q^{98}+101q^{96}-76q^{94}+20q^{92}+47q^{90}-112q^{88}+160q^{86}-165q^{84}+125q^{82}-45q^{80}-57q^{78}+139q^{76}-178q^{74}+161q^{72}-85q^{70}-16q^{68}+105q^{66}-137q^{64}+101q^{62}-3q^{60}-103q^{58}+170q^{56}-158q^{54}+63q^{52}+80q^{50}-214q^{48}+290q^{46}-259q^{44}+139q^{42}+36q^{40}-202q^{38}+298q^{36}-296q^{34}+199q^{32}-49q^{30}-102q^{28}+199q^{26}-211q^{24}+142q^{22}-22q^{20}-97q^{18}+151q^{16}-137q^{14}+42q^{12}+80q^{10}-175q^{8}+207q^{6}-153q^{4}+34q^{2}+104-206q^{-2}+234q^{-4}-184q^{-6}+82q^{-8}+34q^{-10}-121q^{-12}+160q^{-14}-140q^{-16}+92q^{-18}-25q^{-20}-26q^{-22}+52q^{-24}-58q^{-26}+45q^{-28}-26q^{-30}+10q^{-32}+4q^{-34}-9q^{-36}+8q^{-38}-7q^{-40}+4q^{-42}-2q^{-44}+q^{-46}$

Further Quantum Invariants

K11a7 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["K11a7"];

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]=

$q^{3}-3q^{2}+6q-9+13q^{-1}-15q^{-2}+15q^{-3}-13q^{-4}+10q^{-5}-6q^{-6}+3q^{-7}-q^{-8}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{4}z^{10}+a^{2}z^{10}+3a^{5}z^{9}+7a^{3}z^{9}+4az^{9}+4a^{6}z^{8}+7a^{4}z^{8}+8a^{2}z^{8}+5z^{8}+4a^{7}z^{7}-2a^{5}z^{7}-19a^{3}z^{7}-10az^{7}+3z^{7}a^{-1}+3a^{8}z^{6}-4a^{6}z^{6}-26a^{4}z^{6}-37a^{2}z^{6}+z^{6}a^{-2}-17z^{6}+a^{9}z^{5}-5a^{7}z^{5}+a^{5}z^{5}+21a^{3}z^{5}+5az^{5}-9z^{5}a^{-1}-6a^{8}z^{4}+a^{6}z^{4}+41a^{4}z^{4}+55a^{2}z^{4}-3z^{4}a^{-2}+18z^{4}-2a^{9}z^{3}-a^{7}z^{3}-2a^{5}z^{3}-6a^{3}z^{3}+2az^{3}+5z^{3}a^{-1}+3a^{8}z^{2}-4a^{6}z^{2}-27a^{4}z^{2}-31a^{2}z^{2}+z^{2}a^{-2}-10z^{2}+a^{9}z+a^{7}z-az-za^{-1}+2a^{6}+6a^{4}+6a^{2}+3$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_116, K11a33, K11a82,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a325,}

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["K11a7"];

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]=

{ $-t^{4}+5t^{3}-12t^{2}+19t-21+19t^{-1}-12t^{-2}+5t^{-3}-t^{-4}$, $q^{3}-3q^{2}+6q-9+13q^{-1}-15q^{-2}+15q^{-3}-13q^{-4}+10q^{-5}-6q^{-6}+3q^{-7}-q^{-8}$ }

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, K11a33, 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]=

{K11a325,}

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 $0$ $-16$ $0$ $64$ $16$ $0$ $-{\frac {160}{3}}$ ${\frac {32}{3}}$ $16$ $0$ $128$ $0$ $0$ $128$ ${\frac {560}{3}}$ $-{\frac {976}{3}}$ ${\frac {128}{3}}$ $-64$

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 K11a7. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11a6 K11a8