© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11a21 K11a23

Knotscape

This page is passe. Go here instead!    The Knot K11a22 Visit K11a22's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X4251 X8493 X12,5,13,6 X2837 X18,10,19,9 X14,11,15,12 X6,13,7,14 X20,16,21,15 X22,18,1,17 X10,20,11,19 X16,22,17,21

Gauss Code:

{1, -4, 2, -1, 3, -7, 4, -2, 5, -10, 6, -3, 7, -6, 8, -11, 9, -5, 10, -8, 11, -9}

DT (Dowker-Thistlethwaite) Code:

4 8 12 2 18 14 6 20 22 10 16

Alexander Polynomial:

t-4 - 5t-3 + 13t-2 - 20t-1 + 23 - 20t + 13t2 - 5t3 + t4

Conway Polynomial:

1 + 3z2 + 3z4 + 3z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{101, 4}

Jones Polynomial:

- q-1 + 3 - 6q + 11q2 - 13q3 + 16q4 - 16q5 + 14q6 - 11q7 + 6q8 - 3q9 + q10

Other knots (up to mirrors) with the same Jones Polynomial:

{...}

A2 (sl(3)) Invariant:

- q-2 + 1 - 2q2 + q4 + 2q6 + 6q10 - q12 + 3q14 - q16 - 3q18 - 4q22 + q24 + q30

HOMFLY-PT Polynomial:

2a-8 + 3a-8z2 + a-8z4 - 8a-6 - 14a-6z2 - 9a-6z4 - 2a-6z6 + 9a-4 + 19a-4z2 + 15a-4z4 + 6a-4z6 + a-4z8 - 2a-2 - 5a-2z2 - 4a-2z4 - a-2z6

Kauffman Polynomial:

- a-12z2 + a-12z4 + a-11z - 3a-11z3 + 3a-11z5 + a-10z2 - 4a-10z4 + 5a-10z6 - a-9z + 6a-9z3 - 9a-9z5 + 7a-9z7 + 2a-8 - 3a-8z2 + 6a-8z4 - 10a-8z6 + 7a-8z8 - 11a-7z + 29a-7z3 - 26a-7z5 + 3a-7z7 + 4a-7z9 + 8a-6 - 25a-6z2 + 43a-6z4 - 41a-6z6 + 12a-6z8 + a-6z10 - 14a-5z + 32a-5z3 - 17a-5z5 - 11a-5z7 + 7a-5z9 + 9a-4 - 29a-4z2 + 48a-4z4 - 38a-4z6 + 8a-4z8 + a-4z10 - 7a-3z + 17a-3z3 - 7a-3z5 - 6a-3z7 + 3a-3z9 + 2a-2 - 9a-2z2 + 16a-2z4 - 12a-2z6 + 3a-2z8 - 2a-1z + 5a-1z3 - 4a-1z5 + a-1z7

V2 and V3, the type 2 and 3 Vassiliev invariants:

{3, 3}

Khovanov Homology: (The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=4 is the signature of 1122. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 21                       1 j = 19                     2   j = 17                   4 1   j = 15                 7 2     j = 13               7 4       j = 11             9 7         j = 9           7 7           j = 7         6 9             j = 5       5 7               j = 3     2 7                 j = 1   1 4                   j = -1   2                     j = -3 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Show[DrawMorseLink[Knot[11, Alternating, 22]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 22]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[12, 5, 13, 6], X[2, 8, 3, 7], > X[18, 10, 19, 9], X[14, 11, 15, 12], X[6, 13, 7, 14], X[20, 16, 21, 15], > X[22, 18, 1, 17], X[10, 20, 11, 19], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 22]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -7, 4, -2, 5, -10, 6, -3, 7, -6, 8, -11, 9, -5, 10, > -8, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 22]]
Out[5]=	DTCode[4, 8, 12, 2, 18, 14, 6, 20, 22, 10, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 22]][t]
Out[6]=	-4 5 13 20 2 3 4 23 + t - -- + -- - -- - 20 t + 13 t - 5 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 22]][z]
Out[7]=	2 4 6 8 1 + 3 z + 3 z + 3 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 22]}
In[9]:=	{KnotDet[Knot[11, Alternating, 22]], KnotSignature[Knot[11, Alternating, 22]]}
Out[9]=	{101, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 22]][q]
Out[10]=	1 2 3 4 5 6 7 8 9 10 3 - - - 6 q + 11 q - 13 q + 16 q - 16 q + 14 q - 11 q + 6 q - 3 q + q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 22]}
In[12]:=	A2Invariant[Knot[11, Alternating, 22]][q]
Out[12]=	-2 2 4 6 10 12 14 16 18 22 24 1 - q - 2 q + q + 2 q + 6 q - q + 3 q - q - 3 q - 4 q + q + 30 > q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 22]][a, z]
Out[13]=	2 2 2 2 4 4 4 4 2 8 9 2 3 z 14 z 19 z 5 z z 9 z 15 z 4 z -- - -- + -- - -- + ---- - ----- + ----- - ---- + -- - ---- + ----- - ---- - 8 6 4 2 8 6 4 2 8 6 4 2 a a a a a a a a a a a a 6 6 6 8 2 z 6 z z z > ---- + ---- - -- + -- 6 4 2 4 a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 22]][a, z]
Out[14]=	2 2 2 2 8 9 2 z z 11 z 14 z 7 z 2 z z z 3 z -- + -- + -- + -- + --- - -- - ---- - ---- - --- - --- - --- + --- - ---- - 8 6 4 2 11 9 7 5 3 a 12 10 8 a a a a a a a a a a a a 2 2 2 3 3 3 3 3 3 4 25 z 29 z 9 z 3 z 6 z 29 z 32 z 17 z 5 z z > ----- - ----- - ---- - ---- + ---- + ----- + ----- + ----- + ---- + --- - 6 4 2 11 9 7 5 3 a 12 a a a a a a a a a 4 4 4 4 4 5 5 5 5 5 4 z 6 z 43 z 48 z 16 z 3 z 9 z 26 z 17 z 7 z > ---- + ---- + ----- + ----- + ----- + ---- - ---- - ----- - ----- - ---- - 10 8 6 4 2 11 9 7 5 3 a a a a a a a a a a 5 6 6 6 6 6 7 7 7 7 4 z 5 z 10 z 41 z 38 z 12 z 7 z 3 z 11 z 6 z > ---- + ---- - ----- - ----- - ----- - ----- + ---- + ---- - ----- - ---- + a 10 8 6 4 2 9 7 5 3 a a a a a a a a a 7 8 8 8 8 9 9 9 10 10 z 7 z 12 z 8 z 3 z 4 z 7 z 3 z z z > -- + ---- + ----- + ---- + ---- + ---- + ---- + ---- + --- + --- a 8 6 4 2 7 5 3 6 4 a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 22]], Vassiliev[3][Knot[11, Alternating, 22]]}
Out[15]=	{3, 3}
In[16]:=	Kh[Knot[11, Alternating, 22]][q, t]
Out[16]=	3 3 5 1 2 q 4 q 2 q 5 7 7 2 7 q + 5 q + ----- + ---- + -- + --- + ---- + 7 q t + 6 q t + 9 q t + 3 3 2 2 t t q t q t t 9 2 9 3 11 3 11 4 13 4 13 5 15 5 > 7 q t + 7 q t + 9 q t + 7 q t + 7 q t + 4 q t + 7 q t + 15 6 17 6 17 7 19 7 21 8 > 2 q t + 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a22

K11a21 K11a23