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

K11a186 K11a188

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 14 16 20 2 22 8 6 10 18

Alexander Polynomial:

- 2t-3 + 11t-2 - 27t-1 + 37 - 27t + 11t2 - 2t3

Conway Polynomial:

1 - z2 - z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a8, K11a38, K11a249, ...}

Determinant and Signature:

{117, 0}

Jones Polynomial:

q-6 - 3q-5 + 6q-4 - 11q-3 + 16q-2 - 18q-1 + 19 - 17q + 13q2 - 8q3 + 4q4 - q5

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

{...}

A2 (sl(3)) Invariant:

q-18 - q-16 + q-14 + q-12 - 4q-10 + 3q-8 - q-6 - q-4 + 3q-2 - 2 + 4q2 - 2q4 + 2q8 - 3q10 + 2q12 + q14 - q16

HOMFLY-PT Polynomial:

- a-4z2 + 2a-2z2 + 2a-2z4 + 2 + z2 - z4 - z6 - 2a2 - 5a2z2 - 3a2z4 - a2z6 + a4 + 2a4z2 + a4z4

Kauffman Polynomial:

- a-5z3 + a-5z5 + 2a-4z2 - 6a-4z4 + 4a-4z6 - a-3z + 5a-3z3 - 11a-3z5 + 7a-3z7 + 2a-2z2 - 2a-2z4 - 7a-2z6 + 7a-2z8 - 4a-1z + 16a-1z3 - 20a-1z5 + 5a-1z7 + 4a-1z9 + 2 - 8z2 + 20z4 - 26z6 + 12z8 + z10 - 6az + 17az3 - 12az5 - 5az7 + 7az9 + 2a2 - 13a2z2 + 25a2z4 - 25a2z6 + 9a2z8 + a2z10 - 5a3z + 15a3z3 - 13a3z5 + 3a3z9 + a4 - 3a4z2 + 6a4z4 - 9a4z6 + 4a4z8 - 2a5z + 8a5z3 - 9a5z5 + 3a5z7 + 2a6z2 - 3a6z4 + a6z6

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

{-1, 1}

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=0 is the signature of 11187. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 11                       1 j = 9                     3   j = 7                   5 1   j = 5                 8 3     j = 3               9 5       j = 1             10 8         j = -1           9 10           j = -3         7 9             j = -5       4 9               j = -7     2 7                 j = -9   1 4                   j = -11   2                     j = -13 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, 187]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 187]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 4, 13, 3], X[14, 5, 15, 6], X[16, 7, 17, 8], > X[20, 10, 21, 9], X[2, 12, 3, 11], X[22, 13, 1, 14], X[8, 15, 9, 16], > X[6, 17, 7, 18], X[10, 20, 11, 19], X[18, 21, 19, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 187]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -9, 4, -8, 5, -10, 6, -2, 7, -3, 8, -4, 9, -11, 10, > -5, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 187]]
Out[5]=	DTCode[4, 12, 14, 16, 20, 2, 22, 8, 6, 10, 18]
In[6]:=	alex = Alexander[Knot[11, Alternating, 187]][t]
Out[6]=	2 11 27 2 3 37 - -- + -- - -- - 27 t + 11 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 187]][z]
Out[7]=	2 4 6 1 - z - z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 8], Knot[11, Alternating, 38], > Knot[11, Alternating, 187], Knot[11, Alternating, 249]}
In[9]:=	{KnotDet[Knot[11, Alternating, 187]], KnotSignature[Knot[11, Alternating, 187]]}
Out[9]=	{117, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 187]][q]
Out[10]=	-6 3 6 11 16 18 2 3 4 5 19 + q - -- + -- - -- + -- - -- - 17 q + 13 q - 8 q + 4 q - q 5 4 3 2 q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 187]}
In[12]:=	A2Invariant[Knot[11, Alternating, 187]][q]
Out[12]=	-18 -16 -14 -12 4 3 -6 -4 3 2 4 -2 + q - q + q + q - --- + -- - q - q + -- + 4 q - 2 q + 10 8 2 q q q 8 10 12 14 16 > 2 q - 3 q + 2 q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 187]][a, z]
Out[13]=	2 2 4 2 4 2 z 2 z 2 2 4 2 4 2 z 2 4 2 - 2 a + a + z - -- + ---- - 5 a z + 2 a z - z + ---- - 3 a z + 4 2 2 a a a 4 4 6 2 6 > a z - z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 187]][a, z]
Out[14]=	2 2 2 4 z 4 z 3 5 2 2 z 2 z 2 + 2 a + a - -- - --- - 6 a z - 5 a z - 2 a z - 8 z + ---- + ---- - 3 a 4 2 a a a 3 3 3 2 2 4 2 6 2 z 5 z 16 z 3 3 3 > 13 a z - 3 a z + 2 a z - -- + ---- + ----- + 17 a z + 15 a z + 5 3 a a a 4 4 5 5 5 3 4 6 z 2 z 2 4 4 4 6 4 z 11 z > 8 a z + 20 z - ---- - ---- + 25 a z + 6 a z - 3 a z + -- - ----- - 4 2 5 3 a a a a 5 6 6 20 z 5 3 5 5 5 6 4 z 7 z 2 6 > ----- - 12 a z - 13 a z - 9 a z - 26 z + ---- - ---- - 25 a z - a 4 2 a a 7 7 8 4 6 6 6 7 z 5 z 7 5 7 8 7 z 2 8 > 9 a z + a z + ---- + ---- - 5 a z + 3 a z + 12 z + ---- + 9 a z + 3 a 2 a a 9 4 8 4 z 9 3 9 10 2 10 > 4 a z + ---- + 7 a z + 3 a z + z + a z a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 187]], Vassiliev[3][Knot[11, Alternating, 187]]}
Out[15]=	{-1, 1}
In[16]:=	Kh[Knot[11, Alternating, 187]][q, t]
Out[16]=	10 1 2 1 4 2 7 4 9 -- + 10 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 7 9 9 3 3 2 5 2 5 3 > ----- + ---- + --- + 8 q t + 9 q t + 5 q t + 8 q t + 3 q t + 3 2 3 q t q t q t 7 3 7 4 9 4 11 5 > 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a187

K11a186 K11a188