K11a1

10_165

K11a2

	Visit K11a1's page at Knotilus! Visit K11a1's page at the original Knot Atlas!
	K11a1 Quick Notes

K11a1.

A graph, Knot K11a1.

Knot presentations

Planar diagram presentation	X4251 X8394 X10,6,11,5 X14,7,15,8 X2,9,3,10 X16,12,17,11 X20,14,21,13 X6,15,7,16 X22,18,1,17 X12,20,13,19 X18,22,19,21
Gauss code	1, -5, 2, -1, 3, -8, 4, -2, 5, -3, 6, -10, 7, -4, 8, -6, 9, -11, 10, -7, 11, -9
Dowker-Thistlethwaite code	4 8 10 14 2 16 20 6 22 12 18
Conway Notation	[221,211,2]

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a1/ThurstonBennequinNumber Hyperbolic Volume 15.6439 A-Polynomial See Data:K11a1/A-polynomial

[edit Notes for K11a1's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant -2

[edit Notes for K11a1's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-12t^{2}+30t-39+30t^{-1}-12t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 127, 2 }
Jones polynomial	$-q^{8}+4q^{7}-9q^{6}+14q^{5}-18q^{4}+21q^{3}-20q^{2}+17q-12+7q^{-1}-3q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+z^{6}a^{-4}+z^{4}a^{-2}+2z^{4}a^{-4}-z^{4}a^{-6}-2z^{4}+a^{2}z^{2}+3z^{2}a^{-4}-z^{2}a^{-6}-3z^{2}+a^{2}+2a^{-4}-a^{-6}-1$
Kauffman polynomial (db, data sources)	$z^{10}a^{-2}+z^{10}a^{-4}+3z^{9}a^{-1}+8z^{9}a^{-3}+5z^{9}a^{-5}+11z^{8}a^{-2}+16z^{8}a^{-4}+9z^{8}a^{-6}+4z^{8}+3az^{7}+3z^{7}a^{-1}-4z^{7}a^{-3}+4z^{7}a^{-5}+8z^{7}a^{-7}+a^{2}z^{6}-25z^{6}a^{-2}-36z^{6}a^{-4}-14z^{6}a^{-6}+4z^{6}a^{-8}-6z^{6}-8az^{5}-18z^{5}a^{-1}-16z^{5}a^{-3}-20z^{5}a^{-5}-13z^{5}a^{-7}+z^{5}a^{-9}-3a^{2}z^{4}+15z^{4}a^{-2}+28z^{4}a^{-4}+9z^{4}a^{-6}-5z^{4}a^{-8}-2z^{4}+7az^{3}+15z^{3}a^{-1}+16z^{3}a^{-3}+16z^{3}a^{-5}+7z^{3}a^{-7}-z^{3}a^{-9}+3a^{2}z^{2}-3z^{2}a^{-2}-10z^{2}a^{-4}-4z^{2}a^{-6}+z^{2}a^{-8}+5z^{2}-2az-4za^{-1}-4za^{-3}-4za^{-5}-2za^{-7}-a^{2}+2a^{-4}+a^{-6}-1$
The A2 invariant	Data:K11a1/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a1/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a1 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["K11a1"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-12t^{2}+30t-39+30t^{-1}-12t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$2z^{6}+1$

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]=

{ 127, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{8}+4q^{7}-9q^{6}+14q^{5}-18q^{4}+21q^{3}-20q^{2}+17q-12+7q^{-1}-3q^{-2}+q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-2}+z^{10}a^{-4}+3z^{9}a^{-1}+8z^{9}a^{-3}+5z^{9}a^{-5}+11z^{8}a^{-2}+16z^{8}a^{-4}+9z^{8}a^{-6}+4z^{8}+3az^{7}+3z^{7}a^{-1}-4z^{7}a^{-3}+4z^{7}a^{-5}+8z^{7}a^{-7}+a^{2}z^{6}-25z^{6}a^{-2}-36z^{6}a^{-4}-14z^{6}a^{-6}+4z^{6}a^{-8}-6z^{6}-8az^{5}-18z^{5}a^{-1}-16z^{5}a^{-3}-20z^{5}a^{-5}-13z^{5}a^{-7}+z^{5}a^{-9}-3a^{2}z^{4}+15z^{4}a^{-2}+28z^{4}a^{-4}+9z^{4}a^{-6}-5z^{4}a^{-8}-2z^{4}+7az^{3}+15z^{3}a^{-1}+16z^{3}a^{-3}+16z^{3}a^{-5}+7z^{3}a^{-7}-z^{3}a^{-9}+3a^{2}z^{2}-3z^{2}a^{-2}-10z^{2}a^{-4}-4z^{2}a^{-6}+z^{2}a^{-8}+5z^{2}-2az-4za^{-1}-4za^{-3}-4za^{-5}-2za^{-7}-a^{2}+2a^{-4}+a^{-6}-1$

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 $0$ $16$ $0$ $32$ $0$ $0$ ${\frac {64}{3}}$ $-{\frac {128}{3}}$ $16$ $0$ $128$ $0$ $0$ $208$ $-{\frac {544}{3}}$ $224$ $-16$ $48$

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 K11a1. 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                   6 1   -5 11                 8 3     5 9               10 6       -4 7             11 8         3 5           9 10           1 3         8 11             -3 1       5 10               5 -1     2 7                 -5 -3   1 5                   4 -5   2                     -2 -7 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{8}$ $r=1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=3$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[11, Alternating, 1]]
Out[2]=	11
In[3]:=	PD[Knot[11, Alternating, 1]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[10, 6, 11, 5], X[14, 7, 15, 8], X[2, 9, 3, 10], X[16, 12, 17, 11], X[20, 14, 21, 13], X[6, 15, 7, 16], X[22, 18, 1, 17], X[12, 20, 13, 19], X[18, 22, 19, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 1]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -8, 4, -2, 5, -3, 6, -10, 7, -4, 8, -6, 9, -11, 10, -7, 11, -9]
In[5]:=	BR[Knot[11, Alternating, 1]]
Out[5]=	BR[Knot[11, Alternating, 1]]
In[6]:=	alex = Alexander[Knot[11, Alternating, 1]][t]
Out[6]=	2 12 30 -39 + ------------ - ------------ + ----------- + 30 Alternating - 3 2 Alternating Alternating Alternating 2 3 12 Alternating + 2 Alternating
In[7]:=	Conway[Knot[11, Alternating, 1]][z]
Out[7]=	6 1 + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 1], Knot[11, Alternating, 122], Knot[11, Alternating, 149]}
In[9]:=	{KnotDet[Knot[11, Alternating, 1]], KnotSignature[Knot[11, Alternating, 1]]}
Out[9]=	{127, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 1]][q]
Out[10]=	-3 3 7 2 3 4 5 6 -12 + q - -- + - + 17 q - 20 q + 21 q - 18 q + 14 q - 9 q + 2 q q 7 8 4 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 1], Knot[11, Alternating, 149]}
In[12]:=	A2Invariant[Knot[11, Alternating, 1]][q]
Out[12]=	-10 -6 3 2 2 4 6 8 10 12 -1 + q - q + -- - -- + 3 q - 4 q + 3 q - q + q + 3 q - 4 2 q q 14 16 18 20 22 24 3 q + 4 q - 2 q - 2 q + 2 q - q
In[13]:=	Kauffman[Knot[11, Alternating, 1]][a, z]
Out[13]=	2 2 -6 2 2 2 z 4 z 4 z 4 z 2 z 4 z -1 + a + -- - a - --- - --- - --- - --- - 2 a z + 5 z + -- - ---- - 4 7 5 3 a 8 6 a a a a a a 2 2 3 3 3 3 3 10 z 3 z 2 2 z 7 z 16 z 16 z 15 z 3 ----- - ---- + 3 a z - -- + ---- + ----- + ----- + ----- + 7 a z - 4 2 9 7 5 3 a a a a a a a 4 4 4 4 5 5 5 4 5 z 9 z 28 z 15 z 2 4 z 13 z 20 z 2 z - ---- + ---- + ----- + ----- - 3 a z + -- - ----- - ----- - 8 6 4 2 9 7 5 a a a a a a a 5 5 6 6 6 6 16 z 18 z 5 6 4 z 14 z 36 z 25 z ----- - ----- - 8 a z - 6 z + ---- - ----- - ----- - ----- + 3 a 8 6 4 2 a a a a a 7 7 7 7 8 8 2 6 8 z 4 z 4 z 3 z 7 8 9 z 16 z a z + ---- + ---- - ---- + ---- + 3 a z + 4 z + ---- + ----- + 7 5 3 a 6 4 a a a a a 8 9 9 9 10 10 11 z 5 z 8 z 3 z z z ----- + ---- + ---- + ---- + --- + --- 2 5 3 a 4 2 a a a a a
In[14]:=	{Vassiliev[2][Knot[11, Alternating, 1]], Vassiliev[3][Knot[11, Alternating, 1]]}
Out[14]=	{0, 2}
In[15]:=	Kh[Knot[11, Alternating, 1]][q, t]
Out[15]=	1 2 1 5 --------------- + --------------- + --------------- + --------------- + 4 7 3 5 3 3 2 3 Alternating q Alternating q Alternating q Alternating q 2 7 5 q 3 -------------- + ------------- + 10 q + ----------- + 8 q + 2 Alternating q Alternating Alternating q 3 5 2 5 11 Alternating q + 9 Alternating q + 10 Alternating q + 2 7 3 7 3 9 11 Alternating q + 8 Alternating q + 10 Alternating q + 4 9 4 11 5 11 6 Alternating q + 8 Alternating q + 3 Alternating q + 5 13 6 13 6 15 6 Alternating q + Alternating q + 3 Alternating q + 7 17 Alternating q