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

K11a136 K11a138

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 16 14 18 2 20 6 22 12 8

Alexander Polynomial:

2t-3 - 12t-2 + 26t-1 - 31 + 26t - 12t2 + 2t3

Conway Polynomial:

1 - 4z2 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a202, ...}

Determinant and Signature:

{111, -2}

Jones Polynomial:

q-7 - 3q-6 + 7q-5 - 11q-4 + 15q-3 - 18q-2 + 17q-1 - 15 + 12q - 7q2 + 4q3 - q4

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

{...}

A2 (sl(3)) Invariant:

q-22 - q-18 + 3q-16 - q-14 + 2q-10 - 4q-8 + q-6 - 3q-4 + 2 - 2q2 + 4q4 + 2q10 - q12

HOMFLY-PT Polynomial:

a-2 - a-2z2 - a-2z4 + 1 + z2 + 2z4 + z6 - 2a2 - 2a2z2 + a2z4 + a2z6 - 3a4z2 - 2a4z4 + a6 + a6z2

Kauffman Polynomial:

2a-3z3 - 3a-3z5 + a-3z7 - a-2 - 4a-2z2 + 16a-2z4 - 15a-2z6 + 4a-2z8 + 2a-1z - 2a-1z3 + 12a-1z5 - 16a-1z7 + 5a-1z9 + 1 - 17z2 + 44z4 - 36z6 + 5z8 + 2z10 + 6az - 14az3 + 27az5 - 32az7 + 11az9 + 2a2 - 19a2z2 + 40a2z4 - 38a2z6 + 9a2z8 + 2a2z10 + 2a3z - 2a3z5 - 7a3z7 + 6a3z9 + 4a4z4 - 11a4z6 + 8a4z8 - 2a5z + 8a5z3 - 11a5z5 + 8a5z7 - a6 + 5a6z2 - 7a6z4 + 6a6z6 - 2a7z3 + 3a7z5 - a8z2 + a8z4

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

{-4, 4}

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 11137. 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 = 9                       1 j = 7                     3   j = 5                   4 1   j = 3                 8 3     j = 1               7 4       j = -1             10 8         j = -3           9 8           j = -5         6 9             j = -7       5 9               j = -9     2 6                 j = -11   1 5                   j = -13   2                     j = -15 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, 137]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 137]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[16, 6, 17, 5], X[14, 8, 15, 7], > X[18, 9, 19, 10], X[2, 11, 3, 12], X[20, 13, 21, 14], X[6, 16, 7, 15], > X[22, 18, 1, 17], X[12, 19, 13, 20], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 137]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -8, 4, -11, 5, -2, 6, -10, 7, -4, 8, -3, 9, -5, 10, > -7, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 137]]
Out[5]=	DTCode[4, 10, 16, 14, 18, 2, 20, 6, 22, 12, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 137]][t]
Out[6]=	2 12 26 2 3 -31 + -- - -- + -- + 26 t - 12 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 137]][z]
Out[7]=	2 6 1 - 4 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 137], Knot[11, Alternating, 202]}
In[9]:=	{KnotDet[Knot[11, Alternating, 137]], KnotSignature[Knot[11, Alternating, 137]]}
Out[9]=	{111, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 137]][q]
Out[10]=	-7 3 7 11 15 18 17 2 3 4 -15 + q - -- + -- - -- + -- - -- + -- + 12 q - 7 q + 4 q - q 6 5 4 3 2 q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 137]}
In[12]:=	A2Invariant[Knot[11, Alternating, 137]][q]
Out[12]=	-22 -18 3 -14 2 4 -6 3 2 4 10 12 2 + q - q + --- - q + --- - -- + q - -- - 2 q + 4 q + 2 q - q 16 10 8 4 q q q q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 137]][a, z]
Out[13]=	2 4 -2 2 6 2 z 2 2 4 2 6 2 4 z 2 4 1 + a - 2 a + a + z - -- - 2 a z - 3 a z + a z + 2 z - -- + a z - 2 2 a a 4 4 6 2 6 > 2 a z + z + a z
In[14]:=	Kauffman[Knot[11, Alternating, 137]][a, z]
Out[14]=	2 -2 2 6 2 z 3 5 2 4 z 2 2 1 - a + 2 a - a + --- + 6 a z + 2 a z - 2 a z - 17 z - ---- - 19 a z + a 2 a 3 3 6 2 8 2 2 z 2 z 3 5 3 7 3 4 > 5 a z - a z + ---- - ---- - 14 a z + 8 a z - 2 a z + 44 z + 3 a a 4 5 5 16 z 2 4 4 4 6 4 8 4 3 z 12 z 5 > ----- + 40 a z + 4 a z - 7 a z + a z - ---- + ----- + 27 a z - 2 3 a a a 6 3 5 5 5 7 5 6 15 z 2 6 4 6 > 2 a z - 11 a z + 3 a z - 36 z - ----- - 38 a z - 11 a z + 2 a 7 7 8 6 6 z 16 z 7 3 7 5 7 8 4 z > 6 a z + -- - ----- - 32 a z - 7 a z + 8 a z + 5 z + ---- + 3 a 2 a a 9 2 8 4 8 5 z 9 3 9 10 2 10 > 9 a z + 8 a z + ---- + 11 a z + 6 a z + 2 z + 2 a z a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 137]], Vassiliev[3][Knot[11, Alternating, 137]]}
Out[15]=	{-4, 4}
In[16]:=	Kh[Knot[11, Alternating, 137]][q, t]
Out[16]=	8 10 1 2 1 5 2 6 5 9 -- + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 3 q 15 6 13 5 11 5 11 4 9 4 9 3 7 3 7 2 q q t q t q t q t q t q t q t q t 6 9 9 8 t 2 3 2 3 3 5 3 > ----- + ---- + ---- + --- + 7 q t + 4 q t + 8 q t + 3 q t + 4 q t + 5 2 5 3 q q t q t q t 5 4 7 4 9 5 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a137

K11a136 K11a138