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

K11a135 K11a137

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 16 14 18 2 8 20 22 12 6

Alexander Polynomial:

2t-3 - 14t-2 + 39t-1 - 53 + 39t - 14t2 + 2t3

Conway Polynomial:

1 + z2 - 2z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{163, -2}

Jones Polynomial:

q-9 - 4q-8 + 9q-7 - 16q-6 + 22q-5 - 26q-4 + 27q-3 - 23q-2 + 18q-1 - 11 + 5q - q2

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

{...}

A2 (sl(3)) Invariant:

q-28 - q-26 - 2q-24 + 4q-22 - 3q-20 - q-18 + 4q-16 - 5q-14 + 3q-12 - 2q-10 + q-8 + 4q-6 - 4q-4 + 6q-2 - 2 - 2q2 + 3q4 - q6

HOMFLY-PT Polynomial:

- z4 + 2a2 + 2a2z2 + a2z4 + a2z6 - a4 - a4z2 + a4z6 - a6z2 - 2a6z4 + a8z2

Kauffman Polynomial:

a-1z5 - 4z4 + 5z6 + 3az3 - 14az5 + 11az7 - 2a2 + 2a2z2 + 6a2z4 - 20a2z6 + 14a2z8 + 7a3z3 - 12a3z5 - 5a3z7 + 10a3z9 - a4 - a4z2 + 28a4z4 - 47a4z6 + 19a4z8 + 3a4z10 - 2a5z + 6a5z3 + 7a5z5 - 28a5z7 + 17a5z9 - 6a6z2 + 28a6z4 - 37a6z6 + 12a6z8 + 3a6z10 - 3a7z + 8a7z3 - 5a7z5 - 8a7z7 + 7a7z9 - 2a8z2 + 8a8z4 - 14a8z6 + 7a8z8 - a9z + 6a9z3 - 9a9z5 + 4a9z7 + a10z2 - 2a10z4 + a10z6

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=-2 is the signature of 11136. 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 = 5                       1 j = 3                     4   j = 1                   7 1   j = -1                 11 4     j = -3               13 8       j = -5             14 10         j = -7           12 13           j = -9         10 14             j = -11       6 12               j = -13     3 10                 j = -15   1 6                   j = -17   3                     j = -19 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, 136]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 136]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[16, 6, 17, 5], X[14, 7, 15, 8], > X[18, 9, 19, 10], X[2, 11, 3, 12], X[8, 13, 9, 14], X[20, 15, 21, 16], > X[22, 18, 1, 17], X[12, 19, 13, 20], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 136]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -11, 4, -7, 5, -2, 6, -10, 7, -4, 8, -3, 9, -5, 10, > -8, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 136]]
Out[5]=	DTCode[4, 10, 16, 14, 18, 2, 8, 20, 22, 12, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 136]][t]
Out[6]=	2 14 39 2 3 -53 + -- - -- + -- + 39 t - 14 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 136]][z]
Out[7]=	2 4 6 1 + z - 2 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 136]}
In[9]:=	{KnotDet[Knot[11, Alternating, 136]], KnotSignature[Knot[11, Alternating, 136]]}
Out[9]=	{163, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 136]][q]
Out[10]=	-9 4 9 16 22 26 27 23 18 2 -11 + q - -- + -- - -- + -- - -- + -- - -- + -- + 5 q - q 8 7 6 5 4 3 2 q 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, 136]}
In[12]:=	A2Invariant[Knot[11, Alternating, 136]][q]
Out[12]=	-28 -26 2 4 3 -18 4 5 3 2 -8 4 -2 + q - q - --- + --- - --- - q + --- - --- + --- - --- + q + -- - 24 22 20 16 14 12 10 6 q q q q q q q q 4 6 2 4 6 > -- + -- - 2 q + 3 q - q 4 2 q q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 136]][a, z]
Out[13]=	2 4 2 2 4 2 6 2 8 2 4 2 4 6 4 2 6 2 a - a + 2 a z - a z - a z + a z - z + a z - 2 a z + a z + 4 6 > a z
In[14]:=	Kauffman[Knot[11, Alternating, 136]][a, z]
Out[14]=	2 4 5 7 9 2 2 4 2 6 2 8 2 -2 a - a - 2 a z - 3 a z - a z + 2 a z - a z - 6 a z - 2 a z + 10 2 3 3 3 5 3 7 3 9 3 4 2 4 > a z + 3 a z + 7 a z + 6 a z + 8 a z + 6 a z - 4 z + 6 a z + 5 4 4 6 4 8 4 10 4 z 5 3 5 > 28 a z + 28 a z + 8 a z - 2 a z + -- - 14 a z - 12 a z + a 5 5 7 5 9 5 6 2 6 4 6 6 6 > 7 a z - 5 a z - 9 a z + 5 z - 20 a z - 47 a z - 37 a z - 8 6 10 6 7 3 7 5 7 7 7 9 7 > 14 a z + a z + 11 a z - 5 a z - 28 a z - 8 a z + 4 a z + 2 8 4 8 6 8 8 8 3 9 5 9 7 9 > 14 a z + 19 a z + 12 a z + 7 a z + 10 a z + 17 a z + 7 a z + 4 10 6 10 > 3 a z + 3 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 136]], Vassiliev[3][Knot[11, Alternating, 136]]}
Out[15]=	{1, -1}
In[16]:=	Kh[Knot[11, Alternating, 136]][q, t]
Out[16]=	8 11 1 3 1 6 3 10 6 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 19 8 17 7 15 7 15 6 13 6 13 5 11 5 q q t q t q t q t q t q t q t 12 10 14 12 13 14 10 13 4 t > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + --- + 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t q t q t 2 3 2 5 3 > 7 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a136

K11a135 K11a137