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

K11a319 K11a321

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X12,4,13,3 X16,6,17,5 X22,8,1,7 X18,10,19,9 X4,12,5,11 X20,14,21,13 X2,16,3,15 X10,18,11,17 X8,20,9,19 X14,22,15,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 12 16 22 18 4 20 2 10 8 14

Alexander Polynomial:

9t-2 - 27t-1 + 37 - 27t + 9t2

Conway Polynomial:

1 + 9z2 + 9z4

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{109, 4}

Jones Polynomial:

q2 - 3q3 + 7q4 - 11q5 + 16q6 - 17q7 + 17q8 - 15q9 + 11q10 - 7q11 + 3q12 - q13

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

{...}

A2 (sl(3)) Invariant:

q6 - 2q8 + 2q10 + q12 - 3q14 + 4q16 + q20 + 3q22 - q24 + 3q26 - 3q28 - q30 + q32 - 4q34 + q36 + q38 - q40

HOMFLY-PT Polynomial:

- a-12z2 - 3a-10 - 2a-10z2 + a-10z4 + 3a-8 + 7a-8z2 + 4a-8z4 + a-6 + 4a-6z2 + 3a-6z4 + a-4z2 + a-4z4

Kauffman Polynomial:

- 2a-15z + 5a-15z3 - 4a-15z5 + a-15z7 - 4a-14z2 + 12a-14z4 - 11a-14z6 + 3a-14z8 + a-13z - a-13z3 + 8a-13z5 - 12a-13z7 + 4a-13z9 - 4a-12z2 + 20a-12z4 - 21a-12z6 + 2a-12z8 + 2a-12z10 - a-11z - 2a-11z3 + 17a-11z5 - 27a-11z7 + 10a-11z9 + 3a-10 - 15a-10z2 + 32a-10z4 - 33a-10z6 + 8a-10z8 + 2a-10z10 - 4a-9z + 11a-9z3 - 8a-9z5 - 6a-9z7 + 6a-9z9 + 3a-8 - 9a-8z2 + 16a-8z4 - 17a-8z6 + 9a-8z8 + 5a-7z3 - 10a-7z5 + 8a-7z7 - a-6 + 5a-6z2 - 7a-6z4 + 6a-6z6 - 2a-5z3 + 3a-5z5 - a-4z2 + a-4z4

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

{9, 26}

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 11320. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 r = 10 r = 11 j = 27                       1 j = 25                     2   j = 23                   5 1   j = 21                 6 2     j = 19               9 5       j = 17             8 6         j = 15           9 9           j = 13         7 8             j = 11       4 9               j = 9     3 7                 j = 7     4                   j = 5 1 3                     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, 320]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 320]]
Out[3]=	PD[X[6, 2, 7, 1], X[12, 4, 13, 3], X[16, 6, 17, 5], X[22, 8, 1, 7], > X[18, 10, 19, 9], X[4, 12, 5, 11], X[20, 14, 21, 13], X[2, 16, 3, 15], > X[10, 18, 11, 17], X[8, 20, 9, 19], X[14, 22, 15, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 320]]
Out[4]=	GaussCode[1, -8, 2, -6, 3, -1, 4, -10, 5, -9, 6, -2, 7, -11, 8, -3, 9, -5, 10, > -7, 11, -4]
In[5]:=	DTCode[Knot[11, Alternating, 320]]
Out[5]=	DTCode[6, 12, 16, 22, 18, 4, 20, 2, 10, 8, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 320]][t]
Out[6]=	9 27 2 37 + -- - -- - 27 t + 9 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 320]][z]
Out[7]=	2 4 1 + 9 z + 9 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 320]}
In[9]:=	{KnotDet[Knot[11, Alternating, 320]], KnotSignature[Knot[11, Alternating, 320]]}
Out[9]=	{109, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 320]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 11 q - 3 q + 7 q - 11 q + 16 q - 17 q + 17 q - 15 q + 11 q - 7 q + 12 13 > 3 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 320]}
In[12]:=	A2Invariant[Knot[11, Alternating, 320]][q]
Out[12]=	6 8 10 12 14 16 20 22 24 26 28 q - 2 q + 2 q + q - 3 q + 4 q + q + 3 q - q + 3 q - 3 q - 30 32 34 36 38 40 > q + q - 4 q + q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 320]][a, z]
Out[13]=	2 2 2 2 2 4 4 4 4 -3 3 -6 z 2 z 7 z 4 z z z 4 z 3 z z --- + -- + a - --- - ---- + ---- + ---- + -- + --- + ---- + ---- + -- 10 8 12 10 8 6 4 10 8 6 4 a a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 320]][a, z]
Out[14]=	2 2 2 2 2 3 3 -6 2 z z z 4 z 4 z 4 z 15 z 9 z 5 z --- + -- - a - --- + --- - --- - --- - ---- - ---- - ----- - ---- + ---- - 10 8 15 13 11 9 14 12 10 8 6 a a a a a a a a a a a 2 3 3 3 3 3 3 4 4 4 z 5 z z 2 z 11 z 5 z 2 z 12 z 20 z 32 z > -- + ---- - --- - ---- + ----- + ---- - ---- + ----- + ----- + ----- + 4 15 13 11 9 7 5 14 12 10 a a a a a a a a a a 4 4 4 5 5 5 5 5 5 6 16 z 7 z z 4 z 8 z 17 z 8 z 10 z 3 z 11 z > ----- - ---- + -- - ---- + ---- + ----- - ---- - ----- + ---- - ----- - 8 6 4 15 13 11 9 7 5 14 a a a a a a a a a a 6 6 6 6 7 7 7 7 7 8 21 z 33 z 17 z 6 z z 12 z 27 z 6 z 8 z 3 z > ----- - ----- - ----- + ---- + --- - ----- - ----- - ---- + ---- + ---- + 12 10 8 6 15 13 11 9 7 14 a a a a a a a a a a 8 8 8 9 9 9 10 10 2 z 8 z 9 z 4 z 10 z 6 z 2 z 2 z > ---- + ---- + ---- + ---- + ----- + ---- + ----- + ----- 12 10 8 13 11 9 12 10 a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 320]], Vassiliev[3][Knot[11, Alternating, 320]]}
Out[15]=	{9, 26}
In[16]:=	Kh[Knot[11, Alternating, 320]][q, t]
Out[16]=	3 5 5 7 2 9 2 9 3 11 3 11 4 q + q + 3 q t + 4 q t + 3 q t + 7 q t + 4 q t + 9 q t + 13 4 13 5 15 5 15 6 17 6 17 7 > 7 q t + 8 q t + 9 q t + 9 q t + 8 q t + 6 q t + 19 7 19 8 21 8 21 9 23 9 23 10 > 9 q t + 5 q t + 6 q t + 2 q t + 5 q t + q t + 25 10 27 11 > 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a320

K11a319 K11a321