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

K11a189 K11a191

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 14 18 16 2 22 20 8 6 10

Alexander Polynomial:

- 2t-3 + 9t-2 - 19t-1 + 25 - 19t + 9t2 - 2t3

Conway Polynomial:

1 - z2 - 3z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{1086, ...}

Determinant and Signature:

{85, 0}

Jones Polynomial:

q-4 - 3q-3 + 6q-2 - 9q-1 + 12 - 13q + 13q2 - 11q3 + 8q4 - 5q5 + 3q6 - q7

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

{...}

A2 (sl(3)) Invariant:

q-12 - q-10 + q-8 + q-6 - 2q-4 + 3q-2 - 1 - 2q6 + 2q8 - q10 + q12 + 2q14 - q16 + q18 - q22

HOMFLY-PT Polynomial:

- a-6 - a-6z2 + 3a-4 + 5a-4z2 + 2a-4z4 - 2a-2 - 4a-2z2 - 3a-2z4 - a-2z6 - 3z2 - 3z4 - z6 + a2 + 2a2z2 + a2z4

Kauffman Polynomial:

- a-7z + 4a-7z3 - 4a-7z5 + a-7z7 + a-6 - 6a-6z2 + 16a-6z4 - 13a-6z6 + 3a-6z8 - 2a-5z + 3a-5z3 + 6a-5z5 - 10a-5z7 + 3a-5z9 + 3a-4 - 18a-4z2 + 35a-4z4 - 26a-4z6 + 4a-4z8 + a-4z10 - 2a-3z + 11a-3z5 - 17a-3z7 + 6a-3z9 + 2a-2 - 14a-2z2 + 27a-2z4 - 24a-2z6 + 6a-2z8 + a-2z10 - 3a-1z + 11a-1z3 - 11a-1z5 + 3a-1z9 + 2z2 + z4 - 6z6 + 5z8 - 2az + 7az3 - 9az5 + 6az7 - a2 + 3a2z2 - 6a2z4 + 5a2z6 - 3a3z3 + 3a3z5 - a4z2 + a4z4

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 11190. 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 = 15                       1 j = 13                     2   j = 11                   3 1   j = 9                 5 2     j = 7               6 3       j = 5             7 5         j = 3           6 6           j = 1         6 7             j = -1       4 7               j = -3     2 5                 j = -5   1 4                   j = -7   2                     j = -9 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, 190]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 190]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 4, 13, 3], X[14, 5, 15, 6], X[18, 8, 19, 7], > X[16, 10, 17, 9], X[2, 12, 3, 11], X[22, 13, 1, 14], X[20, 15, 21, 16], > X[8, 18, 9, 17], X[6, 20, 7, 19], X[10, 21, 11, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 190]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -10, 4, -9, 5, -11, 6, -2, 7, -3, 8, -5, 9, -4, 10, > -8, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 190]]
Out[5]=	DTCode[4, 12, 14, 18, 16, 2, 22, 20, 8, 6, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 190]][t]
Out[6]=	2 9 19 2 3 25 - -- + -- - -- - 19 t + 9 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 190]][z]
Out[7]=	2 4 6 1 - z - 3 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 86], Knot[11, Alternating, 190]}
In[9]:=	{KnotDet[Knot[11, Alternating, 190]], KnotSignature[Knot[11, Alternating, 190]]}
Out[9]=	{85, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 190]][q]
Out[10]=	-4 3 6 9 2 3 4 5 6 7 12 + q - -- + -- - - - 13 q + 13 q - 11 q + 8 q - 5 q + 3 q - q 3 2 q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 190]}
In[12]:=	A2Invariant[Knot[11, Alternating, 190]][q]
Out[12]=	-12 -10 -8 -6 2 3 6 8 10 12 14 -1 + q - q + q + q - -- + -- - 2 q + 2 q - q + q + 2 q - 4 2 q q 16 18 22 > q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 190]][a, z]
Out[13]=	2 2 2 4 4 -6 3 2 2 2 z 5 z 4 z 2 2 4 2 z 3 z -a + -- - -- + a - 3 z - -- + ---- - ---- + 2 a z - 3 z + ---- - ---- + 4 2 6 4 2 4 2 a a a a a a a 6 2 4 6 z > a z - z - -- 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 190]][a, z]
Out[14]=	2 2 -6 3 2 2 z 2 z 2 z 3 z 2 6 z 18 z a + -- + -- - a - -- - --- - --- - --- - 2 a z + 2 z - ---- - ----- - 4 2 7 5 3 a 6 4 a a a a a a a 2 3 3 3 14 z 2 2 4 2 4 z 3 z 11 z 3 3 3 4 > ----- + 3 a z - a z + ---- + ---- + ----- + 7 a z - 3 a z + z + 2 7 5 a a a a 4 4 4 5 5 5 5 16 z 35 z 27 z 2 4 4 4 4 z 6 z 11 z 11 z > ----- + ----- + ----- - 6 a z + a z - ---- + ---- + ----- - ----- - 6 4 2 7 5 3 a a a a a a a 6 6 6 7 7 5 3 5 6 13 z 26 z 24 z 2 6 z 10 z > 9 a z + 3 a z - 6 z - ----- - ----- - ----- + 5 a z + -- - ----- - 6 4 2 7 5 a a a a a 7 8 8 8 9 9 9 10 10 17 z 7 8 3 z 4 z 6 z 3 z 6 z 3 z z z > ----- + 6 a z + 5 z + ---- + ---- + ---- + ---- + ---- + ---- + --- + --- 3 6 4 2 5 3 a 4 2 a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 190]], Vassiliev[3][Knot[11, Alternating, 190]]}
Out[15]=	{-1, 1}
In[16]:=	Kh[Knot[11, Alternating, 190]][q, t]
Out[16]=	7 1 2 1 4 2 5 4 3 - + 6 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 7 q t + 6 q t + q 9 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 2 5 2 5 3 7 3 7 4 9 4 9 5 > 6 q t + 7 q t + 5 q t + 6 q t + 3 q t + 5 q t + 2 q t + 11 5 11 6 13 6 15 7 > 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a190

K11a189 K11a191