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

K11a300 K11a302

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X10,3,11,4 X18,5,19,6 X14,7,15,8 X16,10,17,9 X4,11,5,12 X22,13,1,14 X20,16,21,15 X12,18,13,17 X2,19,3,20 X8,21,9,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 10 18 14 16 4 22 20 12 2 8

Alexander Polynomial:

- t-4 + 7t-3 - 22t-2 + 43t-1 - 53 + 43t - 22t2 + 7t3 - t4

Conway Polynomial:

1 + 2z2 - z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{199, -2}

Jones Polynomial:

- q-8 + 5q-7 - 13q-6 + 21q-5 - 28q-4 + 33q-3 - 32q-2 + 28q-1 - 20 + 12q - 5q2 + q3

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

{...}

A2 (sl(3)) Invariant:

- q-24 + 2q-22 - q-20 - 4q-18 + 5q-16 - 5q-14 + 3q-12 + 2q-10 - 3q-8 + 7q-6 - 6q-4 + 6q-2 - 1 - 3q2 + 4q4 - 3q6 + q8

HOMFLY-PT Polynomial:

z2 + 2z4 + z6 + a2 - 2a2z2 - 6a2z4 - 4a2z6 - a2z8 + a4 + 4a4z2 + 5a4z4 + 2a4z6 - a6 - a6z2 - a6z4

Kauffman Polynomial:

- a-2z4 + a-2z6 + 3a-1z3 - 8a-1z5 + 5a-1z7 - 3z2 + 14z4 - 22z6 + 11z8 + 4az3 + 2az5 - 18az7 + 12az9 - a2 - 8a2z2 + 43a2z4 - 59a2z6 + 18a2z8 + 5a2z10 + 5a3z3 + 13a3z5 - 47a3z7 + 27a3z9 + a4 - 10a4z2 + 46a4z4 - 70a4z6 + 26a4z8 + 5a4z10 - 2a5z + 10a5z3 - 13a5z5 - 11a5z7 + 15a5z9 + a6 - 5a6z2 + 16a6z4 - 29a6z6 + 19a6z8 - 2a7z + 6a7z3 - 15a7z5 + 13a7z7 - 2a8z4 + 5a8z6 + a9z5

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

{2, -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=-2 is the signature of 11301. 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                     4   j = 3                   8 1   j = 1                 12 4     j = -1               16 8       j = -3             17 13         j = -5           16 15           j = -7         12 17             j = -9       9 16               j = -11     4 12                 j = -13   1 9                   j = -15   4                     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, 301]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 301]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 3, 11, 4], X[18, 5, 19, 6], X[14, 7, 15, 8], > X[16, 10, 17, 9], X[4, 11, 5, 12], X[22, 13, 1, 14], X[20, 16, 21, 15], > X[12, 18, 13, 17], X[2, 19, 3, 20], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 301]]
Out[4]=	GaussCode[1, -10, 2, -6, 3, -1, 4, -11, 5, -2, 6, -9, 7, -4, 8, -5, 9, -3, 10, > -8, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 301]]
Out[5]=	DTCode[6, 10, 18, 14, 16, 4, 22, 20, 12, 2, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 301]][t]
Out[6]=	-4 7 22 43 2 3 4 -53 - t + -- - -- + -- + 43 t - 22 t + 7 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 301]][z]
Out[7]=	2 6 8 1 + 2 z - z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 301]}
In[9]:=	{KnotDet[Knot[11, Alternating, 301]], KnotSignature[Knot[11, Alternating, 301]]}
Out[9]=	{199, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 301]][q]
Out[10]=	-8 5 13 21 28 33 32 28 2 3 -20 - q + -- - -- + -- - -- + -- - -- + -- + 12 q - 5 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, 301]}
In[12]:=	A2Invariant[Knot[11, Alternating, 301]][q]
Out[12]=	-24 2 -20 4 5 5 3 2 3 7 6 6 -1 - q + --- - q - --- + --- - --- + --- + --- - -- + -- - -- + -- - 22 18 16 14 12 10 8 6 4 2 q q q q q q q q q q 2 4 6 8 > 3 q + 4 q - 3 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 301]][a, z]
Out[13]=	2 4 6 2 2 2 4 2 6 2 4 2 4 4 4 a + a - a + z - 2 a z + 4 a z - a z + 2 z - 6 a z + 5 a z - 6 4 6 2 6 4 6 2 8 > a z + z - 4 a z + 2 a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 301]][a, z]
Out[14]=	3 2 4 6 5 7 2 2 2 4 2 6 2 3 z -a + a + a - 2 a z - 2 a z - 3 z - 8 a z - 10 a z - 5 a z + ---- + a 4 3 3 3 5 3 7 3 4 z 2 4 4 4 > 4 a z + 5 a z + 10 a z + 6 a z + 14 z - -- + 43 a z + 46 a z + 2 a 5 6 4 8 4 8 z 5 3 5 5 5 7 5 > 16 a z - 2 a z - ---- + 2 a z + 13 a z - 13 a z - 15 a z + a 6 7 9 5 6 z 2 6 4 6 6 6 8 6 5 z > a z - 22 z + -- - 59 a z - 70 a z - 29 a z + 5 a z + ---- - 2 a a 7 3 7 5 7 7 7 8 2 8 4 8 > 18 a z - 47 a z - 11 a z + 13 a z + 11 z + 18 a z + 26 a z + 6 8 9 3 9 5 9 2 10 4 10 > 19 a z + 12 a z + 27 a z + 15 a z + 5 a z + 5 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 301]], Vassiliev[3][Knot[11, Alternating, 301]]}
Out[15]=	{2, -3}
In[16]:=	Kh[Knot[11, Alternating, 301]][q, t]
Out[16]=	13 16 1 4 1 9 4 12 9 16 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 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 12 17 16 15 17 8 t 2 3 2 > ----- + ----- + ----- + ---- + ---- + --- + 12 q t + 4 q t + 8 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 + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a301

K11a300 K11a302