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

K11a194 K11a196

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 14 20 18 16 2 22 10 8 6

Alexander Polynomial:

3t-2 - 13t-1 + 21 - 13t + 3t2

Conway Polynomial:

1 - z2 + 3z4

Other knots with the same Alexander/Conway Polynomial:

{K11n114, ...}

Determinant and Signature:

{53, 0}

Jones Polynomial:

q-8 - 2q-7 + 3q-6 - 5q-5 + 6q-4 - 7q-3 + 8q-2 - 7q-1 + 6 - 4q + 3q2 - q3

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

{...}

A2 (sl(3)) Invariant:

q-26 + q-24 - q-22 - q-18 - 2q-16 + q-14 + q-8 + q-4 + 2q2 - q4 + q6 + q8 - q10

HOMFLY-PT Polynomial:

- a-2z2 + 1 + z2 + z4 + a2z4 + a4 + a4z2 + a4z4 - 2a6 - 2a6z2 + a8

Kauffman Polynomial:

a-3z3 - 2a-2z2 + 3a-2z4 - 3a-1z3 + 4a-1z5 + 1 - 2z2 - 3z4 + 4z6 - 6az5 + 4az7 - 3a2z2 + 8a2z4 - 11a2z6 + 4a2z8 + 3a3z - 6a3z3 + 9a3z5 - 10a3z7 + 3a3z9 + a4 - 12a4z2 + 29a4z4 - 19a4z6 + a4z8 + a4z10 + 7a5z - 27a5z3 + 42a5z5 - 26a5z7 + 5a5z9 + 2a6 - 16a6z2 + 26a6z4 - 10a6z6 - 2a6z8 + a6z10 + 4a7z - 17a7z3 + 23a7z5 - 12a7z7 + 2a7z9 + a8 - 7a8z2 + 11a8z4 - 6a8z6 + a8z8

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

{-1, 3}

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 11195. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 7                       1 j = 5                     2   j = 3                   2 1   j = 1                 4 2     j = -1               4 3       j = -3             4 3         j = -5           3 4           j = -7         3 4             j = -9       2 3               j = -11     1 3                 j = -13   1 2                   j = -15   1                     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, 195]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 195]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 3, 13, 4], X[14, 6, 15, 5], X[20, 7, 21, 8], > X[18, 9, 19, 10], X[16, 11, 17, 12], X[2, 13, 3, 14], X[22, 16, 1, 15], > X[10, 17, 11, 18], X[8, 19, 9, 20], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 195]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -11, 4, -10, 5, -9, 6, -2, 7, -3, 8, -6, 9, -5, 10, > -4, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 195]]
Out[5]=	DTCode[4, 12, 14, 20, 18, 16, 2, 22, 10, 8, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 195]][t]
Out[6]=	3 13 2 21 + -- - -- - 13 t + 3 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 195]][z]
Out[7]=	2 4 1 - z + 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 195], Knot[11, NonAlternating, 114]}
In[9]:=	{KnotDet[Knot[11, Alternating, 195]], KnotSignature[Knot[11, Alternating, 195]]}
Out[9]=	{53, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 195]][q]
Out[10]=	-8 2 3 5 6 7 8 7 2 3 6 + q - -- + -- - -- + -- - -- + -- - - - 4 q + 3 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, 195]}
In[12]:=	A2Invariant[Knot[11, Alternating, 195]][q]
Out[12]=	-26 -24 -22 -18 2 -14 -8 -4 2 4 6 8 10 q + q - q - q - --- + q + q + q + 2 q - q + q + q - q 16 q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 195]][a, z]
Out[13]=	2 4 6 8 2 z 4 2 6 2 4 2 4 4 4 1 + a - 2 a + a + z - -- + a z - 2 a z + z + a z + a z 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 195]][a, z]
Out[14]=	2 4 6 8 3 5 7 2 2 z 2 2 1 + a + 2 a + a + 3 a z + 7 a z + 4 a z - 2 z - ---- - 3 a z - 2 a 3 3 4 2 6 2 8 2 z 3 z 3 3 5 3 7 3 > 12 a z - 16 a z - 7 a z + -- - ---- - 6 a z - 27 a z - 17 a z - 3 a a 4 5 4 3 z 2 4 4 4 6 4 8 4 4 z 5 > 3 z + ---- + 8 a z + 29 a z + 26 a z + 11 a z + ---- - 6 a z + 2 a a 3 5 5 5 7 5 6 2 6 4 6 6 6 > 9 a z + 42 a z + 23 a z + 4 z - 11 a z - 19 a z - 10 a z - 8 6 7 3 7 5 7 7 7 2 8 4 8 > 6 a z + 4 a z - 10 a z - 26 a z - 12 a z + 4 a z + a z - 6 8 8 8 3 9 5 9 7 9 4 10 6 10 > 2 a z + a z + 3 a z + 5 a z + 2 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 195]], Vassiliev[3][Knot[11, Alternating, 195]]}
Out[15]=	{-1, 3}
In[16]:=	Kh[Knot[11, Alternating, 195]][q, t]
Out[16]=	3 1 1 1 2 1 3 2 3 - + 4 q + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + q 17 8 15 7 13 7 13 6 11 6 11 5 9 5 9 4 q t q t q t q t q t q t q t q t 3 4 3 4 4 3 4 3 > ----- + ----- + ----- + ----- + ----- + ---- + --- + 2 q t + 2 q t + 7 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 7 3 > q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a195

K11a194 K11a196