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

K11a301 K11a303

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 10 18 14 20 4 22 8 12 2 16

Alexander Polynomial:

t-4 - 7t-3 + 20t-2 - 33t-1 + 39 - 33t + 20t2 - 7t3 + t4

Conway Polynomial:

1 - 2z4 + z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{161, -4}

Jones Polynomial:

q-10 - 4q-9 + 10q-8 - 17q-7 + 22q-6 - 26q-5 + 26q-4 - 22q-3 + 17q-2 - 10q-1 + 5 - q

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

{...}

A2 (sl(3)) Invariant:

q-30 - q-28 + 3q-24 - 4q-22 + 3q-20 - 3q-18 - 2q-16 + 3q-14 - 5q-12 + 6q-10 - 3q-8 + 2q-6 + 3q-4 - 2q-2 + 3 - q2

HOMFLY-PT Polynomial:

2a2 + a2z2 - 2a2z4 - a2z6 + 2a4z2 + 5a4z4 + 4a4z6 + a4z8 - 2a6 - 5a6z2 - 6a6z4 - 2a6z6 + a8 + 2a8z2 + a8z4

Kauffman Polynomial:

az3 - 2az5 + az7 - 2a2 - a2z2 + 13a2z4 - 15a2z6 + 5a2z8 + a3z - a3z3 + 17a3z5 - 23a3z7 + 8a3z9 - 11a4z2 + 47a4z4 - 46a4z6 + 6a4z8 + 4a4z10 - a5z + 29a5z5 - 53a5z7 + 21a5z9 + 2a6 - 18a6z2 + 59a6z4 - 73a6z6 + 20a6z8 + 4a6z10 - 5a7z + 16a7z3 - 18a7z5 - 12a7z7 + 13a7z9 + a8 - 5a8z2 + 17a8z4 - 32a8z6 + 19a8z8 - 3a9z + 14a9z3 - 24a9z5 + 17a9z7 + 3a10z2 - 7a10z4 + 10a10z6 + 4a11z5 + a12z4

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

{0, 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=-4 is the signature of 11302. 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 = 3                       1 j = 1                     4   j = -1                   6 1   j = -3                 11 4     j = -5               12 7       j = -7             14 10         j = -9           12 12           j = -11         10 14             j = -13       7 12               j = -15     3 10                 j = -17   1 7                   j = -19   3                     j = -21 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, 302]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 302]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 3, 11, 4], X[18, 5, 19, 6], X[14, 7, 15, 8], > X[20, 10, 21, 9], X[4, 11, 5, 12], X[22, 13, 1, 14], X[8, 15, 9, 16], > X[12, 18, 13, 17], X[2, 19, 3, 20], X[16, 21, 17, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 302]]
Out[4]=	GaussCode[1, -10, 2, -6, 3, -1, 4, -8, 5, -2, 6, -9, 7, -4, 8, -11, 9, -3, 10, > -5, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 302]]
Out[5]=	DTCode[6, 10, 18, 14, 20, 4, 22, 8, 12, 2, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 302]][t]
Out[6]=	-4 7 20 33 2 3 4 39 + t - -- + -- - -- - 33 t + 20 t - 7 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 302]][z]
Out[7]=	4 6 8 1 - 2 z + z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 302]}
In[9]:=	{KnotDet[Knot[11, Alternating, 302]], KnotSignature[Knot[11, Alternating, 302]]}
Out[9]=	{161, -4}
In[10]:=	J=Jones[Knot[11, Alternating, 302]][q]
Out[10]=	-10 4 10 17 22 26 26 22 17 10 5 + q - -- + -- - -- + -- - -- + -- - -- + -- - -- - q 9 8 7 6 5 4 3 2 q 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, 302]}
In[12]:=	A2Invariant[Knot[11, Alternating, 302]][q]
Out[12]=	-30 -28 3 4 3 3 2 3 5 6 3 2 3 + q - q + --- - --- + --- - --- - --- + --- - --- + --- - -- + -- + 24 22 20 18 16 14 12 10 8 6 q q q q q q q q q q 3 2 2 > -- - -- - q 4 2 q q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 302]][a, z]
Out[13]=	2 6 8 2 2 4 2 6 2 8 2 2 4 4 4 2 a - 2 a + a + a z + 2 a z - 5 a z + 2 a z - 2 a z + 5 a z - 6 4 8 4 2 6 4 6 6 6 4 8 > 6 a z + a z - a z + 4 a z - 2 a z + a z
In[14]:=	Kauffman[Knot[11, Alternating, 302]][a, z]
Out[14]=	2 6 8 3 5 7 9 2 2 4 2 -2 a + 2 a + a + a z - a z - 5 a z - 3 a z - a z - 11 a z - 6 2 8 2 10 2 3 3 3 7 3 9 3 > 18 a z - 5 a z + 3 a z + a z - a z + 16 a z + 14 a z + 2 4 4 4 6 4 8 4 10 4 12 4 5 > 13 a z + 47 a z + 59 a z + 17 a z - 7 a z + a z - 2 a z + 3 5 5 5 7 5 9 5 11 5 2 6 > 17 a z + 29 a z - 18 a z - 24 a z + 4 a z - 15 a z - 4 6 6 6 8 6 10 6 7 3 7 5 7 > 46 a z - 73 a z - 32 a z + 10 a z + a z - 23 a z - 53 a z - 7 7 9 7 2 8 4 8 6 8 8 8 3 9 > 12 a z + 17 a z + 5 a z + 6 a z + 20 a z + 19 a z + 8 a z + 5 9 7 9 4 10 6 10 > 21 a z + 13 a z + 4 a z + 4 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 302]], Vassiliev[3][Knot[11, Alternating, 302]]}
Out[15]=	{0, 2}
In[16]:=	Kh[Knot[11, Alternating, 302]][q, t]
Out[16]=	7 11 1 3 1 7 3 10 7 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 21 8 19 7 17 7 17 6 15 6 15 5 13 5 q q q t q t q t q t q t q t q t 12 10 14 12 12 14 10 12 4 t > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + --- + 13 4 11 4 11 3 9 3 9 2 7 2 7 5 3 q t q t q t q t q t q t q t q t q 2 6 t t 2 3 3 > --- + -- + 4 q t + q t q q

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a302

K11a301 K11a303