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

K11a215 K11a217

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X12,4,13,3 X18,6,19,5 X14,7,15,8 X20,10,21,9 X2,12,3,11 X22,13,1,14 X8,15,9,16 X10,18,11,17 X6,20,7,19 X16,21,17,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 18 14 20 2 22 8 10 6 16

Alexander Polynomial:

- t-4 + 6t-3 - 17t-2 + 31t-1 - 37 + 31t - 17t2 + 6t3 - t4

Conway Polynomial:

1 + z2 - z4 - 2z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{K11a196, K11a286, ...}

Determinant and Signature:

{147, 2}

Jones Polynomial:

q-3 - 4q-2 + 9q-1 - 15 + 21q - 23q2 + 24q3 - 21q4 + 15q5 - 9q6 + 4q7 - q8

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

{K11a196, ...}

A2 (sl(3)) Invariant:

q-8 - 2q-6 + 3q-4 - 2q-2 - 1 + 4q2 - 4q4 + 6q6 - 2q8 + q10 + q12 - 4q14 + 4q16 - 2q18 + q22 - q24

HOMFLY-PT Polynomial:

- a-6 - 2a-6z2 - a-6z4 + 2a-4 + 8a-4z2 + 7a-4z4 + 2a-4z6 - a-2 - 8a-2z2 - 10a-2z4 - 5a-2z6 - a-2z8 + 1 + 3z2 + 3z4 + z6

Kauffman Polynomial:

- a-9z3 + a-9z5 + a-8z2 - 5a-8z4 + 4a-8z6 - 2a-7z + 7a-7z3 - 12a-7z5 + 8a-7z7 + a-6 - 4a-6z2 + 10a-6z4 - 15a-6z6 + 10a-6z8 - 4a-5z + 18a-5z3 - 18a-5z5 + 7a-5z9 + 2a-4 - 15a-4z2 + 41a-4z4 - 47a-4z6 + 18a-4z8 + 2a-4z10 - 4a-3z + 16a-3z3 - 9a-3z5 - 16a-3z7 + 13a-3z9 + a-2 - 15a-2z2 + 38a-2z4 - 44a-2z6 + 15a-2z8 + 2a-2z10 - 3a-1z + 12a-1z3 - 13a-1z5 - 4a-1z7 + 6a-1z9 + 1 - 4z2 + 10z4 - 15z6 + 7z8 - az + 6az3 - 9az5 + 4az7 + a2z2 - 2a2z4 + a2z6

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

{1, 2}

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 11216. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 17                       1 j = 15                     3   j = 13                   6 1   j = 11                 9 3     j = 9               12 6       j = 7             12 9         j = 5           11 12           j = 3         10 12             j = 1       6 12               j = -1     3 9                 j = -3   1 6                   j = -5   3                     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, 216]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 216]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 4, 13, 3], X[18, 6, 19, 5], X[14, 7, 15, 8], > X[20, 10, 21, 9], X[2, 12, 3, 11], X[22, 13, 1, 14], X[8, 15, 9, 16], > X[10, 18, 11, 17], X[6, 20, 7, 19], X[16, 21, 17, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 216]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -10, 4, -8, 5, -9, 6, -2, 7, -4, 8, -11, 9, -3, 10, > -5, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 216]]
Out[5]=	DTCode[4, 12, 18, 14, 20, 2, 22, 8, 10, 6, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 216]][t]
Out[6]=	-4 6 17 31 2 3 4 -37 - t + -- - -- + -- + 31 t - 17 t + 6 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 216]][z]
Out[7]=	2 4 6 8 1 + z - z - 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 196], Knot[11, Alternating, 216], > Knot[11, Alternating, 286]}
In[9]:=	{KnotDet[Knot[11, Alternating, 216]], KnotSignature[Knot[11, Alternating, 216]]}
Out[9]=	{147, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 216]][q]
Out[10]=	-3 4 9 2 3 4 5 6 7 8 -15 + q - -- + - + 21 q - 23 q + 24 q - 21 q + 15 q - 9 q + 4 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 196], Knot[11, Alternating, 216]}
In[12]:=	A2Invariant[Knot[11, Alternating, 216]][q]
Out[12]=	-8 2 3 2 2 4 6 8 10 12 14 -1 + q - -- + -- - -- + 4 q - 4 q + 6 q - 2 q + q + q - 4 q + 6 4 2 q q q 16 18 22 24 > 4 q - 2 q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 216]][a, z]
Out[13]=	2 2 2 4 4 4 -6 2 -2 2 2 z 8 z 8 z 4 z 7 z 10 z 1 - a + -- - a + 3 z - ---- + ---- - ---- + 3 z - -- + ---- - ----- + 4 6 4 2 6 4 2 a a a a a a a 6 6 8 6 2 z 5 z z > z + ---- - ---- - -- 4 2 2 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 216]][a, z]
Out[14]=	2 2 2 -6 2 -2 2 z 4 z 4 z 3 z 2 z 4 z 15 z 1 + a + -- + a - --- - --- - --- - --- - a z - 4 z + -- - ---- - ----- - 4 7 5 3 a 8 6 4 a a a a a a a 2 3 3 3 3 3 4 15 z 2 2 z 7 z 18 z 16 z 12 z 3 4 5 z > ----- + a z - -- + ---- + ----- + ----- + ----- + 6 a z + 10 z - ---- + 2 9 7 5 3 a 8 a a a a a a 4 4 4 5 5 5 5 5 10 z 41 z 38 z 2 4 z 12 z 18 z 9 z 13 z > ----- + ----- + ----- - 2 a z + -- - ----- - ----- - ---- - ----- - 6 4 2 9 7 5 3 a a a a a a a a 6 6 6 6 7 7 5 6 4 z 15 z 47 z 44 z 2 6 8 z 16 z > 9 a z - 15 z + ---- - ----- - ----- - ----- + a z + ---- - ----- - 8 6 4 2 7 3 a a a a a a 7 8 8 8 9 9 9 4 z 7 8 10 z 18 z 15 z 7 z 13 z 6 z > ---- + 4 a z + 7 z + ----- + ----- + ----- + ---- + ----- + ---- + a 6 4 2 5 3 a a a a a a 10 10 2 z 2 z > ----- + ----- 4 2 a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 216]], Vassiliev[3][Knot[11, Alternating, 216]]}
Out[15]=	{1, 2}
In[16]:=	Kh[Knot[11, Alternating, 216]][q, t]
Out[16]=	3 1 3 1 6 3 9 6 q 3 12 q + 10 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 12 q t + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 11 q t + 12 q t + 12 q t + 9 q t + 12 q t + 6 q t + 9 q t + 11 5 13 5 13 6 15 6 17 7 > 3 q t + 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a216

K11a215 K11a217