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

K11a362 K11a364

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DrawMorseLink

PD Presentation:

X8291 X16,4,17,3 X14,6,15,5 X18,8,19,7 X22,10,1,9 X20,12,21,11 X6,14,7,13 X4,16,5,15 X2,18,3,17 X12,20,13,19 X10,22,11,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

8 16 14 18 22 20 6 4 2 12 10

Alexander Polynomial:

9t-1 - 17 + 9t

Conway Polynomial:

1 + 9z2

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{35, 2}

Jones Polynomial:

q - 2q2 + 3q3 - 4q4 + 5q5 - 4q6 + 5q7 - 4q8 + 3q9 - 2q10 + q11 - q12

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

{...}

A2 (sl(3)) Invariant:

q2 - q4 + q8 - q10 + q14 + q16 + 2q18 + q20 + q22 - q26 + q28 - q32 - q36 - q38

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

9a-13z - 24a-13z3 + 22a-13z5 - 8a-13z7 + a-13z9 - a-12 + 3a-12z2 - 13a-12z4 + 16a-12z6 - 7a-12z8 + a-12z10 + 9a-11z - 36a-11z3 + 44a-11z5 - 20a-11z7 + 3a-11z9 - 5a-10z2 + 8a-10z4 + a-10z6 - 4a-10z8 + a-10z10 + 2a-9z3 + 6a-9z5 - 8a-9z7 + 2a-9z9 + 4a-8z2 + 4a-8z4 - 10a-8z6 + 3a-8z8 + 6a-7z3 - 12a-7z5 + 4a-7z7 - 2a-6 + 9a-6z2 - 14a-6z4 + 5a-6z6 - 6a-5z3 + 4a-5z5 - 2a-4z2 + 3a-4z4 + 2a-3z3 + a-2z2

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

{9, 27}

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=2 is the signature of 11363. 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 = 25                       1 j = 23                         j = 21                   2 1   j = 19                 1       j = 17               3 2       j = 15             2 1         j = 13           2 3           j = 11         3 2             j = 9       1 2               j = 7     2 3                 j = 5     1                   j = 3 1 2                     j = 1 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, 363]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 363]]
Out[3]=	PD[X[8, 2, 9, 1], X[16, 4, 17, 3], X[14, 6, 15, 5], X[18, 8, 19, 7], > X[22, 10, 1, 9], X[20, 12, 21, 11], X[6, 14, 7, 13], X[4, 16, 5, 15], > X[2, 18, 3, 17], X[12, 20, 13, 19], X[10, 22, 11, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 363]]
Out[4]=	GaussCode[1, -9, 2, -8, 3, -7, 4, -1, 5, -11, 6, -10, 7, -3, 8, -2, 9, -4, 10, > -6, 11, -5]
In[5]:=	DTCode[Knot[11, Alternating, 363]]
Out[5]=	DTCode[8, 16, 14, 18, 22, 20, 6, 4, 2, 12, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 363]][t]
Out[6]=	9 -17 + - + 9 t t
In[7]:=	Conway[Knot[11, Alternating, 363]][z]
Out[7]=	2 1 + 9 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 363]}
In[9]:=	{KnotDet[Knot[11, Alternating, 363]], KnotSignature[Knot[11, Alternating, 363]]}
Out[9]=	{35, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 363]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 11 12 q - 2 q + 3 q - 4 q + 5 q - 4 q + 5 q - 4 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, 363]}
In[12]:=	A2Invariant[Knot[11, Alternating, 363]][q]
Out[12]=	2 4 8 10 14 16 18 20 22 26 28 32 36 38 q - q + q - q + q + q + 2 q + q + q - q + q - q - q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 363]][a, z]
Out[13]=	2 2 2 2 2 -12 2 z 2 z 3 z 2 z z -a + -- + --- + ---- + ---- + ---- + -- 6 10 8 6 4 2 a a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 363]][a, z]
Out[14]=	2 2 2 2 2 2 3 -12 2 9 z 9 z 3 z 5 z 4 z 9 z 2 z z 24 z -a - -- + --- + --- + ---- - ---- + ---- + ---- - ---- + -- - ----- - 6 13 11 12 10 8 6 4 2 13 a a a a a a a a a a 3 3 3 3 3 4 4 4 4 4 36 z 2 z 6 z 6 z 2 z 13 z 8 z 4 z 14 z 3 z > ----- + ---- + ---- - ---- + ---- - ----- + ---- + ---- - ----- + ---- + 11 9 7 5 3 12 10 8 6 4 a a a a a a a a a a 5 5 5 5 5 6 6 6 6 7 22 z 44 z 6 z 12 z 4 z 16 z z 10 z 5 z 8 z > ----- + ----- + ---- - ----- + ---- + ----- + --- - ----- + ---- - ---- - 13 11 9 7 5 12 10 8 6 13 a a a a a a a a a a 7 7 7 8 8 8 9 9 9 10 10 20 z 8 z 4 z 7 z 4 z 3 z z 3 z 2 z z z > ----- - ---- + ---- - ---- - ---- + ---- + --- + ---- + ---- + --- + --- 11 9 7 12 10 8 13 11 9 12 10 a a a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 363]], Vassiliev[3][Knot[11, Alternating, 363]]}
Out[15]=	{9, 27}
In[16]:=	Kh[Knot[11, Alternating, 363]][q, t]
Out[16]=	3 3 5 2 7 2 7 3 9 3 9 4 11 4 q + q + 2 q t + q t + 2 q t + 3 q t + q t + 2 q t + 3 q t + 11 5 13 5 13 6 15 6 15 7 17 7 17 8 > 2 q t + 2 q t + 3 q t + 2 q t + q t + 3 q t + 2 q t + 19 8 21 9 21 10 25 11 > q t + 2 q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a363

K11a362 K11a364