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

K11a363 K11a365

Knotscape

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DrawMorseLink

PD Presentation:

X10,2,11,1 X14,4,15,3 X16,6,17,5 X18,8,19,7 X20,10,21,9 X22,12,1,11 X2,14,3,13 X4,16,5,15 X6,18,7,17 X8,20,9,19 X12,22,13,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

10 14 16 18 20 22 2 4 6 8 12

Alexander Polynomial:

2t-4 - 3t-3 + 3t-2 - 3t-1 + 3 - 3t + 3t2 - 3t3 + 2t4

Conway Polynomial:

1 + 14z2 + 25z4 + 13z6 + 2z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{25, 8}

Jones Polynomial:

q4 - q5 + 2q6 - 2q7 + 3q8 - 3q9 + 3q10 - 3q11 + 3q12 - 2q13 + q14 - q15

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

{...}

A2 (sl(3)) Invariant:

q14 + q18 + q22 + q24 + q26 + q28 + q32 - q38 - q40 - q42 - q44

HOMFLY-PT Polynomial:

- 4a-12 - 10a-12z2 - 6a-12z4 - a-12z6 + 4a-10 + 14a-10z2 + 16a-10z4 + 7a-10z6 + a-10z8 + a-8 + 10a-8z2 + 15a-8z4 + 7a-8z6 + a-8z8

Kauffman Polynomial:

- 2a-19z + a-19z3 - a-18z2 + a-18z4 + a-17z - a-17z3 + a-17z5 + 3a-16z2 - 2a-16z4 + a-16z6 - a-15z + 4a-15z3 - 3a-15z5 + a-15z7 - 4a-14z2 + 6a-14z4 - 4a-14z6 + a-14z8 + a-13z - 8a-13z3 + 9a-13z5 - 5a-13z7 + a-13z9 - 4a-12 + 19a-12z2 - 30a-12z4 + 20a-12z6 - 7a-12z8 + a-12z10 + 5a-11z - 18a-11z3 + 23a-11z5 - 12a-11z7 + 2a-11z9 - 4a-10 + 17a-10z2 - 24a-10z4 + 18a-10z6 - 7a-10z8 + a-10z10 - 4a-9z3 + 10a-9z5 - 6a-9z7 + a-9z9 + a-8 - 10a-8z2 + 15a-8z4 - 7a-8z6 + a-8z8

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

{14, 50}

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=8 is the signature of 11364. 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 = 31                       1 j = 29                         j = 27                   2 1   j = 25                 1       j = 23               2 2       j = 21             1 1         j = 19           2 2           j = 17         1 1             j = 15       1 2               j = 13     1 1                 j = 11     1                   j = 9 1 1                     j = 7 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, 364]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 364]]
Out[3]=	PD[X[10, 2, 11, 1], X[14, 4, 15, 3], X[16, 6, 17, 5], X[18, 8, 19, 7], > X[20, 10, 21, 9], X[22, 12, 1, 11], X[2, 14, 3, 13], X[4, 16, 5, 15], > X[6, 18, 7, 17], X[8, 20, 9, 19], X[12, 22, 13, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 364]]
Out[4]=	GaussCode[1, -7, 2, -8, 3, -9, 4, -10, 5, -1, 6, -11, 7, -2, 8, -3, 9, -4, 10, > -5, 11, -6]
In[5]:=	DTCode[Knot[11, Alternating, 364]]
Out[5]=	DTCode[10, 14, 16, 18, 20, 22, 2, 4, 6, 8, 12]
In[6]:=	alex = Alexander[Knot[11, Alternating, 364]][t]
Out[6]=	2 3 3 3 2 3 4 3 + -- - -- + -- - - - 3 t + 3 t - 3 t + 2 t 4 3 2 t t t t
In[7]:=	Conway[Knot[11, Alternating, 364]][z]
Out[7]=	2 4 6 8 1 + 14 z + 25 z + 13 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 364]}
In[9]:=	{KnotDet[Knot[11, Alternating, 364]], KnotSignature[Knot[11, Alternating, 364]]}
Out[9]=	{25, 8}
In[10]:=	J=Jones[Knot[11, Alternating, 364]][q]
Out[10]=	4 5 6 7 8 9 10 11 12 13 14 15 q - q + 2 q - 2 q + 3 q - 3 q + 3 q - 3 q + 3 q - 2 q + q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 364]}
In[12]:=	A2Invariant[Knot[11, Alternating, 364]][q]
Out[12]=	14 18 22 24 26 28 32 38 40 42 44 q + q + q + q + q + q + q - q - q - q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 364]][a, z]
Out[13]=	2 2 2 4 4 4 6 6 -4 4 -8 10 z 14 z 10 z 6 z 16 z 15 z z 7 z --- + --- + a - ----- + ----- + ----- - ---- + ----- + ----- - --- + ---- + 12 10 12 10 8 12 10 8 12 10 a a a a a a a a a a 6 8 8 7 z z z > ---- + --- + -- 8 10 8 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 364]][a, z]
Out[14]=	2 2 2 2 -4 4 -8 2 z z z z 5 z z 3 z 4 z 19 z --- - --- + a - --- + --- - --- + --- + --- - --- + ---- - ---- + ----- + 12 10 19 17 15 13 11 18 16 14 12 a a a a a a a a a a a 2 2 3 3 3 3 3 3 4 4 17 z 10 z z z 4 z 8 z 18 z 4 z z 2 z > ----- - ----- + --- - --- + ---- - ---- - ----- - ---- + --- - ---- + 10 8 19 17 15 13 11 9 18 16 a a a a a a a a a a 4 4 4 4 5 5 5 5 5 6 6 z 30 z 24 z 15 z z 3 z 9 z 23 z 10 z z > ---- - ----- - ----- + ----- + --- - ---- + ---- + ----- + ----- + --- - 14 12 10 8 17 15 13 11 9 16 a a a a a a a a a a 6 6 6 6 7 7 7 7 8 8 4 z 20 z 18 z 7 z z 5 z 12 z 6 z z 7 z > ---- + ----- + ----- - ---- + --- - ---- - ----- - ---- + --- - ---- - 14 12 10 8 15 13 11 9 14 12 a a a a a a a a a a 8 8 9 9 9 10 10 7 z z z 2 z z z z > ---- + -- + --- + ---- + -- + --- + --- 10 8 13 11 9 12 10 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 364]], Vassiliev[3][Knot[11, Alternating, 364]]}
Out[15]=	{14, 50}
In[16]:=	Kh[Knot[11, Alternating, 364]][q, t]
Out[16]=	7 9 9 11 2 13 2 13 3 15 3 15 4 17 4 q + q + q t + q t + q t + q t + q t + 2 q t + q t + 17 5 19 5 19 6 21 6 21 7 23 7 23 8 > q t + 2 q t + 2 q t + q t + q t + 2 q t + 2 q t + 25 8 27 9 27 10 31 11 > q t + 2 q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a364

K11a363 K11a365