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

K11a35 K11a37

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X14,6,15,5 X2837 X16,9,17,10 X20,11,21,12 X6,14,7,13 X12,15,13,16 X22,17,1,18 X10,19,11,20 X18,21,19,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 14 2 16 20 6 12 22 10 18

Alexander Polynomial:

- 2t-3 + 12t-2 - 28t-1 + 37 - 28t + 12t2 - 2t3

Conway Polynomial:

1 + 2z2 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a169, ...}

Determinant and Signature:

{121, 0}

Jones Polynomial:

q-6 - 4q-5 + 8q-4 - 13q-3 + 17q-2 - 19q-1 + 20 - 16q + 12q2 - 7q3 + 3q4 - q5

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

{K11a35, K11a316, ...}

A2 (sl(3)) Invariant:

q-18 - 2q-16 + q-14 + q-12 - 4q-10 + 3q-8 - 2q-6 + 3q-2 - 1 + 5q2 - 2q4 + q6 + 2q8 - 3q10 + q12 - q16

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

a-5z - 2a-5z3 + a-5z5 - a-4 + 3a-4z2 - 5a-4z4 + 3a-4z6 - a-3z + 4a-3z3 - 7a-3z5 + 5a-3z7 - a-2 + 5a-2z2 - 5a-2z4 - 2a-2z6 + 5a-2z8 - 6a-1z + 20a-1z3 - 23a-1z5 + 8a-1z7 + 3a-1z9 + 2 + 7z4 - 19z6 + 11z8 + z10 - 8az + 25az3 - 25az5 + 7az9 + a2 - 3a2z2 + 15a2z4 - 28a2z6 + 12a2z8 + a2z10 - 5a3z + 18a3z3 - 20a3z5 + a3z7 + 4a3z9 + 6a4z4 - 13a4z6 + 6a4z8 - a5z + 7a5z3 - 10a5z5 + 4a5z7 + a6z2 - 2a6z4 + a6z6

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

{2, 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 1136. 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                     2   j = 7                   5 1   j = 5                 7 2     j = 3               9 5       j = 1             11 7         j = -1           9 10           j = -3         8 10             j = -5       5 9               j = -7     3 8                 j = -9   1 5                   j = -11   3                     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, 36]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 36]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[14, 6, 15, 5], X[2, 8, 3, 7], > X[16, 9, 17, 10], X[20, 11, 21, 12], X[6, 14, 7, 13], X[12, 15, 13, 16], > X[22, 17, 1, 18], X[10, 19, 11, 20], X[18, 21, 19, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 36]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -7, 4, -2, 5, -10, 6, -8, 7, -3, 8, -5, 9, -11, 10, > -6, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 36]]
Out[5]=	DTCode[4, 8, 14, 2, 16, 20, 6, 12, 22, 10, 18]
In[6]:=	alex = Alexander[Knot[11, Alternating, 36]][t]
Out[6]=	2 12 28 2 3 37 - -- + -- - -- - 28 t + 12 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 36]][z]
Out[7]=	2 6 1 + 2 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 36], Knot[11, Alternating, 169]}
In[9]:=	{KnotDet[Knot[11, Alternating, 36]], KnotSignature[Knot[11, Alternating, 36]]}
Out[9]=	{121, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 36]][q]
Out[10]=	-6 4 8 13 17 19 2 3 4 5 20 + q - -- + -- - -- + -- - -- - 16 q + 12 q - 7 q + 3 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, 35], Knot[11, Alternating, 36], > Knot[11, Alternating, 316]}
In[12]:=	A2Invariant[Knot[11, Alternating, 36]][q]
Out[12]=	-18 2 -14 -12 4 3 2 3 2 4 6 8 -1 + q - --- + q + q - --- + -- - -- + -- + 5 q - 2 q + q + 2 q - 16 10 8 6 2 q q q q q 10 12 16 > 3 q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 36]][a, z]
Out[13]=	2 2 4 -4 -2 2 2 z 3 z 2 2 4 2 4 2 z 2 4 2 - a + a - a + z - -- + ---- - 2 a z + a z - z + ---- - 2 a z + 4 2 2 a a a 4 4 6 2 6 > a z - z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 36]][a, z]
Out[14]=	2 2 -4 -2 2 z z 6 z 3 5 3 z 5 z 2 - a - a + a + -- - -- - --- - 8 a z - 5 a z - a z + ---- + ---- - 5 3 a 4 2 a a a a 3 3 3 2 2 6 2 2 z 4 z 20 z 3 3 3 5 3 > 3 a z + a z - ---- + ---- + ----- + 25 a z + 18 a z + 7 a z + 5 3 a a a 4 4 5 5 5 4 5 z 5 z 2 4 4 4 6 4 z 7 z 23 z > 7 z - ---- - ---- + 15 a z + 6 a z - 2 a z + -- - ---- - ----- - 4 2 5 3 a a a a a 6 6 5 3 5 5 5 6 3 z 2 z 2 6 4 6 > 25 a z - 20 a z - 10 a z - 19 z + ---- - ---- - 28 a z - 13 a z + 4 2 a a 7 7 8 6 6 5 z 8 z 3 7 5 7 8 5 z 2 8 4 8 > a z + ---- + ---- + a z + 4 a z + 11 z + ---- + 12 a z + 6 a z + 3 a 2 a a 9 3 z 9 3 9 10 2 10 > ---- + 7 a z + 4 a z + z + a z a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 36]], Vassiliev[3][Knot[11, Alternating, 36]]}
Out[15]=	{2, 1}
In[16]:=	Kh[Knot[11, Alternating, 36]][q, t]
Out[16]=	10 1 3 1 5 3 8 5 9 -- + 11 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 8 10 9 3 3 2 5 2 5 3 > ----- + ---- + --- + 7 q t + 9 q t + 5 q t + 7 q t + 2 q t + 3 2 3 q t q t q t 7 3 7 4 9 4 11 5 > 5 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a36

K11a35 K11a37