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

K11a299 K11a301

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 10 18 14 2 20 22 8 12 4 16

Alexander Polynomial:

t-4 - 6t-3 + 17t-2 - 32t-1 + 41 - 32t + 17t2 - 6t3 + t4

Conway Polynomial:

1 - 2z2 + z4 + 2z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{153, 0}

Jones Polynomial:

q-6 - 4q-5 + 10q-4 - 16q-3 + 21q-2 - 25q-1 + 25 - 21q + 16q2 - 9q3 + 4q4 - q5

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

{...}

A2 (sl(3)) Invariant:

q-18 - q-16 + q-14 + 3q-12 - 4q-10 + 3q-8 - 3q-6 - 2q-4 + 3q-2 - 4 + 6q2 - 3q4 + 2q6 + 3q8 - 3q10 + 2q12 - q14

HOMFLY-PT Polynomial:

- 3a-2z2 - 3a-2z4 - a-2z6 + 3 + 8z2 + 10z4 + 5z6 + z8 - 4a2 - 9a2z2 - 7a2z4 - 2a2z6 + 2a4 + 2a4z2 + a4z4

Kauffman Polynomial:

- a-5z3 + a-5z5 + a-4z2 - 5a-4z4 + 4a-4z6 - a-3z + 4a-3z3 - 11a-3z5 + 8a-3z7 - 6a-2z2 + 15a-2z4 - 19a-2z6 + 11a-2z8 - 2a-1z + 7a-1z3 - a-1z5 - 10a-1z7 + 9a-1z9 + 3 - 23z2 + 57z4 - 53z6 + 16z8 + 3z10 + 20az5 - 35az7 + 17az9 + 4a2 - 25a2z2 + 52a2z4 - 49a2z6 + 13a2z8 + 3a2z10 + 2a3z3 + a3z5 - 13a3z7 + 8a3z9 + 2a4 - 8a4z2 + 13a4z4 - 18a4z6 + 8a4z8 - a5z + 4a5z3 - 8a5z5 + 4a5z7 + a6z2 - 2a6z4 + a6z6

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

{-2, 2}

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 11300. 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                     3   j = 7                   6 1   j = 5                 10 3     j = 3               11 6       j = 1             14 10         j = -1           12 12           j = -3         9 13             j = -5       7 12               j = -7     3 9                 j = -9   1 7                   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, 300]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 300]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 4, 11, 3], X[18, 5, 19, 6], X[14, 7, 15, 8], > X[2, 10, 3, 9], X[20, 12, 21, 11], X[22, 13, 1, 14], X[8, 15, 9, 16], > X[12, 18, 13, 17], X[4, 19, 5, 20], X[16, 21, 17, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 300]]
Out[4]=	GaussCode[1, -5, 2, -10, 3, -1, 4, -8, 5, -2, 6, -9, 7, -4, 8, -11, 9, -3, 10, > -6, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 300]]
Out[5]=	DTCode[6, 10, 18, 14, 2, 20, 22, 8, 12, 4, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 300]][t]
Out[6]=	-4 6 17 32 2 3 4 41 + t - -- + -- - -- - 32 t + 17 t - 6 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 300]][z]
Out[7]=	2 4 6 8 1 - 2 z + z + 2 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 300]}
In[9]:=	{KnotDet[Knot[11, Alternating, 300]], KnotSignature[Knot[11, Alternating, 300]]}
Out[9]=	{153, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 300]][q]
Out[10]=	-6 4 10 16 21 25 2 3 4 5 25 + q - -- + -- - -- + -- - -- - 21 q + 16 q - 9 q + 4 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, 300]}
In[12]:=	A2Invariant[Knot[11, Alternating, 300]][q]
Out[12]=	-18 -16 -14 3 4 3 3 2 3 2 4 6 -4 + q - q + q + --- - --- + -- - -- - -- + -- + 6 q - 3 q + 2 q + 12 10 8 6 4 2 q q q q q q 8 10 12 14 > 3 q - 3 q + 2 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 300]][a, z]
Out[13]=	2 4 2 4 2 3 z 2 2 4 2 4 3 z 2 4 3 - 4 a + 2 a + 8 z - ---- - 9 a z + 2 a z + 10 z - ---- - 7 a z + 2 2 a a 6 4 4 6 z 2 6 8 > a z + 5 z - -- - 2 a z + z 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 300]][a, z]
Out[14]=	2 2 2 4 z 2 z 5 2 z 6 z 2 2 4 2 3 + 4 a + 2 a - -- - --- - a z - 23 z + -- - ---- - 25 a z - 8 a z + 3 a 4 2 a a a 3 3 3 4 4 6 2 z 4 z 7 z 3 3 5 3 4 5 z 15 z > a z - -- + ---- + ---- + 2 a z + 4 a z + 57 z - ---- + ----- + 5 3 a 4 2 a a a a 5 5 5 2 4 4 4 6 4 z 11 z z 5 3 5 > 52 a z + 13 a z - 2 a z + -- - ----- - -- + 20 a z + a z - 5 3 a a a 6 6 7 5 5 6 4 z 19 z 2 6 4 6 6 6 8 z > 8 a z - 53 z + ---- - ----- - 49 a z - 18 a z + a z + ---- - 4 2 3 a a a 7 8 10 z 7 3 7 5 7 8 11 z 2 8 4 8 > ----- - 35 a z - 13 a z + 4 a z + 16 z + ----- + 13 a z + 8 a z + a 2 a 9 9 z 9 3 9 10 2 10 > ---- + 17 a z + 8 a z + 3 z + 3 a z a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 300]], Vassiliev[3][Knot[11, Alternating, 300]]}
Out[15]=	{-2, 2}
In[16]:=	Kh[Knot[11, Alternating, 300]][q, t]
Out[16]=	12 1 3 1 7 3 9 7 12 -- + 14 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 9 13 12 3 3 2 5 2 5 3 > ----- + ---- + --- + 10 q t + 11 q t + 6 q t + 10 q t + 3 q t + 3 2 3 q t q t q t 7 3 7 4 9 4 11 5 > 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a300

K11a299 K11a301