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

K11a326 K11a328

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 12 18 16 14 4 20 22 2 10 8

Alexander Polynomial:

3t-3 - 17t-2 + 44t-1 - 59 + 44t - 17t2 + 3t3

Conway Polynomial:

1 + 3z2 + z4 + 3z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{187, 2}

Jones Polynomial:

- q-2 + 5q-1 - 12 + 20q - 26q2 + 31q3 - 30q4 + 26q5 - 19q6 + 11q7 - 5q8 + q9

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

{...}

A2 (sl(3)) Invariant:

- q-6 + 3q-4 - 2q-2 - 3 + 6q2 - 5q4 + 4q6 + 2q8 - 2q10 + 5q12 - 5q14 + 5q16 - q18 - 4q20 + 4q22 - 3q24 - q26 + q28

HOMFLY-PT Polynomial:

a-8z2 - 2a-6 - 4a-6z2 - 3a-6z4 + 3a-4 + 7a-4z2 + 5a-4z4 + 2a-4z6 - a-2z2 + a-2z6 - z4

Kauffman Polynomial:

- a-10z4 + a-10z6 + 4a-9z3 - 9a-9z5 + 5a-9z7 - 3a-8z2 + 12a-8z4 - 20a-8z6 + 10a-8z8 - 5a-7z + 16a-7z3 - 11a-7z5 - 11a-7z7 + 10a-7z9 + 2a-6 - 15a-6z2 + 50a-6z4 - 61a-6z6 + 19a-6z8 + 4a-6z10 - 8a-5z + 23a-5z3 - 3a-5z5 - 36a-5z7 + 23a-5z9 + 3a-4 - 16a-4z2 + 54a-4z4 - 71a-4z6 + 26a-4z8 + 4a-4z10 - 4a-3z + 15a-3z3 - 16a-3z5 - 8a-3z7 + 13a-3z9 - 4a-2z2 + 14a-2z4 - 26a-2z6 + 17a-2z8 - a-1z + 4a-1z3 - 14a-1z5 + 12a-1z7 - 3z4 + 5z6 + az5

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

{3, 5}

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 11327. 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 = 19                       1 j = 17                     4   j = 15                   7 1   j = 13                 12 4     j = 11               14 7       j = 9             16 12         j = 7           15 14           j = 5         11 16             j = 3       9 15               j = 1     4 12                 j = -1   1 8                   j = -3   4                     j = -5 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, 327]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 327]]
Out[3]=	PD[X[6, 2, 7, 1], X[12, 4, 13, 3], X[18, 6, 19, 5], X[16, 7, 17, 8], > X[14, 10, 15, 9], X[4, 12, 5, 11], X[20, 14, 21, 13], X[22, 15, 1, 16], > X[2, 18, 3, 17], X[10, 20, 11, 19], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 327]]
Out[4]=	GaussCode[1, -9, 2, -6, 3, -1, 4, -11, 5, -10, 6, -2, 7, -5, 8, -4, 9, -3, 10, > -7, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 327]]
Out[5]=	DTCode[6, 12, 18, 16, 14, 4, 20, 22, 2, 10, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 327]][t]
Out[6]=	3 17 44 2 3 -59 + -- - -- + -- + 44 t - 17 t + 3 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 327]][z]
Out[7]=	2 4 6 1 + 3 z + z + 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 327]}
In[9]:=	{KnotDet[Knot[11, Alternating, 327]], KnotSignature[Knot[11, Alternating, 327]]}
Out[9]=	{187, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 327]][q]
Out[10]=	-2 5 2 3 4 5 6 7 8 9 -12 - q + - + 20 q - 26 q + 31 q - 30 q + 26 q - 19 q + 11 q - 5 q + q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 327]}
In[12]:=	A2Invariant[Knot[11, Alternating, 327]][q]
Out[12]=	-6 3 2 2 4 6 8 10 12 14 -3 - q + -- - -- + 6 q - 5 q + 4 q + 2 q - 2 q + 5 q - 5 q + 4 2 q q 16 18 20 22 24 26 28 > 5 q - q - 4 q + 4 q - 3 q - q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 327]][a, z]
Out[13]=	2 2 2 2 4 4 6 6 -2 3 z 4 z 7 z z 4 3 z 5 z 2 z z -- + -- + -- - ---- + ---- - -- - z - ---- + ---- + ---- + -- 6 4 8 6 4 2 6 4 4 2 a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 327]][a, z]
Out[14]=	2 2 2 2 3 3 2 3 5 z 8 z 4 z z 3 z 15 z 16 z 4 z 4 z 16 z -- + -- - --- - --- - --- - - - ---- - ----- - ----- - ---- + ---- + ----- + 6 4 7 5 3 a 8 6 4 2 9 7 a a a a a a a a a a a 3 3 3 4 4 4 4 4 5 23 z 15 z 4 z 4 z 12 z 50 z 54 z 14 z 9 z > ----- + ----- + ---- - 3 z - --- + ----- + ----- + ----- + ----- - ---- - 5 3 a 10 8 6 4 2 9 a a a a a a a a 5 5 5 5 6 6 6 6 11 z 3 z 16 z 14 z 5 6 z 20 z 61 z 71 z > ----- - ---- - ----- - ----- + a z + 5 z + --- - ----- - ----- - ----- - 7 5 3 a 10 8 6 4 a a a a a a a 6 7 7 7 7 7 8 8 8 26 z 5 z 11 z 36 z 8 z 12 z 10 z 19 z 26 z > ----- + ---- - ----- - ----- - ---- + ----- + ----- + ----- + ----- + 2 9 7 5 3 a 8 6 4 a a a a a a a a 8 9 9 9 10 10 17 z 10 z 23 z 13 z 4 z 4 z > ----- + ----- + ----- + ----- + ----- + ----- 2 7 5 3 6 4 a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 327]], Vassiliev[3][Knot[11, Alternating, 327]]}
Out[15]=	{3, 5}
In[16]:=	Kh[Knot[11, Alternating, 327]][q, t]
Out[16]=	3 1 4 1 8 4 q 3 5 5 2 12 q + 9 q + ----- + ----- + ---- + --- + --- + 15 q t + 11 q t + 16 q t + 5 3 3 2 2 q t t q t q t q t 7 2 7 3 9 3 9 4 11 4 11 5 > 15 q t + 14 q t + 16 q t + 12 q t + 14 q t + 7 q t + 13 5 13 6 15 6 15 7 17 7 19 8 > 12 q t + 4 q t + 7 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a327

K11a326 K11a328