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

K11a328 K11a330

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 12 18 22 14 4 20 8 2 10 16

Alexander Polynomial:

11t-2 - 36t-1 + 51 - 36t + 11t2

Conway Polynomial:

1 + 8z2 + 11z4

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{145, 4}

Jones Polynomial:

q2 - 4q3 + 10q4 - 15q5 + 21q6 - 23q7 + 23q8 - 20q9 + 14q10 - 9q11 + 4q12 - q13

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

{...}

A2 (sl(3)) Invariant:

q6 - 3q8 + 3q10 + 2q12 - 4q14 + 5q16 - q18 + q20 + 4q22 - q24 + 4q26 - 5q28 - 2q30 + q32 - 5q34 + 2q36 + 2q38 - q40

HOMFLY-PT Polynomial:

a-12 - a-12z2 - 5a-10 - 4a-10z2 + a-10z4 + 4a-8 + 8a-8z2 + 5a-8z4 + a-6 + 5a-6z2 + 4a-6z4 + a-4z4

Kauffman Polynomial:

- a-15z + 3a-15z3 - 3a-15z5 + a-15z7 - 4a-14z2 + 13a-14z4 - 13a-14z6 + 4a-14z8 - a-13z + 2a-13z3 + 11a-13z5 - 17a-13z7 + 6a-13z9 + a-12 - 8a-12z2 + 36a-12z4 - 36a-12z6 + 5a-12z8 + 3a-12z10 - 7a-11z + 13a-11z3 + 13a-11z5 - 38a-11z7 + 16a-11z9 + 5a-10 - 20a-10z2 + 49a-10z4 - 59a-10z6 + 17a-10z8 + 3a-10z10 - 7a-9z + 21a-9z3 - 24a-9z5 - 5a-9z7 + 10a-9z9 + 4a-8 - 11a-8z2 + 16a-8z4 - 26a-8z6 + 16a-8z8 + 7a-7z3 - 19a-7z5 + 15a-7z7 - a-6 + 5a-6z2 - 9a-6z4 + 10a-6z6 + 4a-5z5 + a-4z4

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

{8, 21}

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 11329. 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                     3   j = 23                   6 1   j = 21                 8 3     j = 19               12 6       j = 17             11 8         j = 15           12 12           j = 13         9 11             j = 11       6 12               j = 9     4 9                 j = 7     6                   j = 5 1 4                     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, 329]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 329]]
Out[3]=	PD[X[6, 2, 7, 1], X[12, 4, 13, 3], X[18, 6, 19, 5], X[22, 8, 1, 7], > X[14, 10, 15, 9], X[4, 12, 5, 11], X[20, 14, 21, 13], X[8, 16, 9, 15], > X[2, 18, 3, 17], X[10, 20, 11, 19], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 329]]
Out[4]=	GaussCode[1, -9, 2, -6, 3, -1, 4, -8, 5, -10, 6, -2, 7, -5, 8, -11, 9, -3, 10, > -7, 11, -4]
In[5]:=	DTCode[Knot[11, Alternating, 329]]
Out[5]=	DTCode[6, 12, 18, 22, 14, 4, 20, 8, 2, 10, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 329]][t]
Out[6]=	11 36 2 51 + -- - -- - 36 t + 11 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 329]][z]
Out[7]=	2 4 1 + 8 z + 11 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 329]}
In[9]:=	{KnotDet[Knot[11, Alternating, 329]], KnotSignature[Knot[11, Alternating, 329]]}
Out[9]=	{145, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 329]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 11 q - 4 q + 10 q - 15 q + 21 q - 23 q + 23 q - 20 q + 14 q - 9 q + 12 13 > 4 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 329]}
In[12]:=	A2Invariant[Knot[11, Alternating, 329]][q]
Out[12]=	6 8 10 12 14 16 18 20 22 24 26 q - 3 q + 3 q + 2 q - 4 q + 5 q - q + q + 4 q - q + 4 q - 28 30 32 34 36 38 40 > 5 q - 2 q + q - 5 q + 2 q + 2 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 329]][a, z]
Out[13]=	2 2 2 2 4 4 4 4 -12 5 4 -6 z 4 z 8 z 5 z z 5 z 4 z z a - --- + -- + a - --- - ---- + ---- + ---- + --- + ---- + ---- + -- 10 8 12 10 8 6 10 8 6 4 a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 329]][a, z]
Out[14]=	2 2 2 2 -12 5 4 -6 z z 7 z 7 z 4 z 8 z 20 z 11 z a + --- + -- - a - --- - --- - --- - --- - ---- - ---- - ----- - ----- + 10 8 15 13 11 9 14 12 10 8 a a a a a a a a a a 2 3 3 3 3 3 4 4 4 4 5 z 3 z 2 z 13 z 21 z 7 z 13 z 36 z 49 z 16 z > ---- + ---- + ---- + ----- + ----- + ---- + ----- + ----- + ----- + ----- - 6 15 13 11 9 7 14 12 10 8 a a a a a a a a a a 4 4 5 5 5 5 5 5 6 6 9 z z 3 z 11 z 13 z 24 z 19 z 4 z 13 z 36 z > ---- + -- - ---- + ----- + ----- - ----- - ----- + ---- - ----- - ----- - 6 4 15 13 11 9 7 5 14 12 a a a a a a a a a a 6 6 6 7 7 7 7 7 8 8 59 z 26 z 10 z z 17 z 38 z 5 z 15 z 4 z 5 z > ----- - ----- + ----- + --- - ----- - ----- - ---- + ----- + ---- + ---- + 10 8 6 15 13 11 9 7 14 12 a a a a a a a a a a 8 8 9 9 9 10 10 17 z 16 z 6 z 16 z 10 z 3 z 3 z > ----- + ----- + ---- + ----- + ----- + ----- + ----- 10 8 13 11 9 12 10 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 329]], Vassiliev[3][Knot[11, Alternating, 329]]}
Out[15]=	{8, 21}
In[16]:=	Kh[Knot[11, Alternating, 329]][q, t]
Out[16]=	3 5 5 7 2 9 2 9 3 11 3 11 4 q + q + 4 q t + 6 q t + 4 q t + 9 q t + 6 q t + 12 q t + 13 4 13 5 15 5 15 6 17 6 17 7 > 9 q t + 11 q t + 12 q t + 12 q t + 11 q t + 8 q t + 19 7 19 8 21 8 21 9 23 9 23 10 > 12 q t + 6 q t + 8 q t + 3 q t + 6 q t + q t + 25 10 27 11 > 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a329

K11a328 K11a330