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

K11a195 K11a197

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 14 20 22 18 2 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:

{K11a216, K11a286, ...}

Determinant and Signature:

{147, -2}

Jones Polynomial:

- q-8 + 4q-7 - 9q-6 + 15q-5 - 21q-4 + 24q-3 - 23q-2 + 21q-1 - 15 + 9q - 4q2 + q3

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

{K11a216, ...}

A2 (sl(3)) Invariant:

- q-24 + q-22 - 2q-18 + 4q-16 - 4q-14 + q-12 + q-10 - 2q-8 + 6q-6 - 4q-4 + 4q-2 - 1 - 2q2 + 3q4 - 2q6 + q8

HOMFLY-PT Polynomial:

1 + 3z2 + 3z4 + z6 - a2 - 8a2z2 - 10a2z4 - 5a2z6 - a2z8 + 2a4 + 8a4z2 + 7a4z4 + 2a4z6 - a6 - 2a6z2 - a6z4

Kauffman Polynomial:

a-2z2 - 2a-2z4 + a-2z6 - a-1z + 6a-1z3 - 9a-1z5 + 4a-1z7 + 1 - 4z2 + 10z4 - 15z6 + 7z8 - 3az + 12az3 - 13az5 - 4az7 + 6az9 + a2 - 15a2z2 + 38a2z4 - 44a2z6 + 15a2z8 + 2a2z10 - 4a3z + 16a3z3 - 9a3z5 - 16a3z7 + 13a3z9 + 2a4 - 15a4z2 + 41a4z4 - 47a4z6 + 18a4z8 + 2a4z10 - 4a5z + 18a5z3 - 18a5z5 + 7a5z9 + a6 - 4a6z2 + 10a6z4 - 15a6z6 + 10a6z8 - 2a7z + 7a7z3 - 12a7z5 + 8a7z7 + a8z2 - 5a8z4 + 4a8z6 - a9z3 + a9z5

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 11196. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

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

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a196

K11a195 K11a197