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

K11a247 K11a249

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X8394 X10,6,11,5 X18,8,19,7 X16,9,17,10 X20,11,21,12 X22,13,1,14 X4,16,5,15 X2,17,3,18 X14,19,15,20 X12,21,13,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 8 10 18 16 20 22 4 2 14 12

Alexander Polynomial:

- t-4 + 6t-3 - 18t-2 + 34t-1 - 41 + 34t - 18t2 + 6t3 - t4

Conway Polynomial:

1 - 2z4 - 2z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{K11a71, ...}

Determinant and Signature:

{159, -2}

Jones Polynomial:

- q-8 + 5q-7 - 11q-6 + 17q-5 - 23q-4 + 26q-3 - 25q-2 + 22q-1 - 15 + 9q - 4q2 + q3

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

{K11a71, ...}

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

1 + 3z2 + 3z4 + z6 - 7a2z2 - 10a2z4 - 5a2z6 - a2z8 + 5a4z2 + 6a4z4 + 2a4z6 - a6z2 - a6z4

Kauffman Polynomial:

- 2a-2z4 + a-2z6 + 4a-1z3 - 9a-1z5 + 4a-1z7 + 1 - 7z2 + 19z4 - 21z6 + 8z8 + az - 2az3 + 10az5 - 16az7 + 8az9 - 15a2z2 + 45a2z4 - 44a2z6 + 12a2z8 + 3a2z10 + 3a3z - 7a3z3 + 21a3z5 - 33a3z7 + 17a3z9 - 9a4z2 + 32a4z4 - 44a4z6 + 17a4z8 + 3a4z10 + 3a5z + 3a5z3 - 14a5z5 - 2a5z7 + 9a5z9 - a6z2 + 4a6z4 - 17a6z6 + 13a6z8 + a7z + 4a7z3 - 15a7z5 + 11a7z7 - 4a8z4 + 5a8z6 + a9z5

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

{0, 0}

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 11248. 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                     3   j = 3                   6 1   j = 1                 9 3     j = -1               13 6       j = -3             13 10         j = -5           13 12           j = -7         10 13             j = -9       7 13               j = -11     4 10                 j = -13   1 7                   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, 248]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 248]]
Out[3]=	PD[X[6, 2, 7, 1], X[8, 3, 9, 4], X[10, 6, 11, 5], X[18, 8, 19, 7], > X[16, 9, 17, 10], X[20, 11, 21, 12], X[22, 13, 1, 14], X[4, 16, 5, 15], > X[2, 17, 3, 18], X[14, 19, 15, 20], X[12, 21, 13, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 248]]
Out[4]=	GaussCode[1, -9, 2, -8, 3, -1, 4, -2, 5, -3, 6, -11, 7, -10, 8, -5, 9, -4, 10, > -6, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 248]]
Out[5]=	DTCode[6, 8, 10, 18, 16, 20, 22, 4, 2, 14, 12]
In[6]:=	alex = Alexander[Knot[11, Alternating, 248]][t]
Out[6]=	-4 6 18 34 2 3 4 -41 - t + -- - -- + -- + 34 t - 18 t + 6 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 248]][z]
Out[7]=	4 6 8 1 - 2 z - 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 71], Knot[11, Alternating, 248]}
In[9]:=	{KnotDet[Knot[11, Alternating, 248]], KnotSignature[Knot[11, Alternating, 248]]}
Out[9]=	{159, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 248]][q]
Out[10]=	-8 5 11 17 23 26 25 22 2 3 -15 - q + -- - -- + -- - -- + -- - -- + -- + 9 q - 4 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, 71], Knot[11, Alternating, 248]}
In[12]:=	A2Invariant[Knot[11, Alternating, 248]][q]
Out[12]=	-24 2 2 4 5 -12 2 6 4 5 2 4 -1 - q + --- - --- + --- - --- + q - -- + -- - -- + -- - 2 q + 3 q - 22 18 16 14 8 6 4 2 q q q q q q q q 6 8 > 2 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 248]][a, z]
Out[13]=	2 2 2 4 2 6 2 4 2 4 4 4 6 4 6 1 + 3 z - 7 a z + 5 a z - a z + 3 z - 10 a z + 6 a z - a z + z - 2 6 4 6 2 8 > 5 a z + 2 a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 248]][a, z]
Out[14]=	3 3 5 7 2 2 2 4 2 6 2 4 z 1 + a z + 3 a z + 3 a z + a z - 7 z - 15 a z - 9 a z - a z + ---- - a 4 3 3 3 5 3 7 3 4 2 z 2 4 4 4 > 2 a z - 7 a z + 3 a z + 4 a z + 19 z - ---- + 45 a z + 32 a z + 2 a 5 6 4 8 4 9 z 5 3 5 5 5 7 5 > 4 a z - 4 a z - ---- + 10 a z + 21 a z - 14 a z - 15 a z + a 6 7 9 5 6 z 2 6 4 6 6 6 8 6 4 z > a z - 21 z + -- - 44 a z - 44 a z - 17 a z + 5 a z + ---- - 2 a a 7 3 7 5 7 7 7 8 2 8 4 8 > 16 a z - 33 a z - 2 a z + 11 a z + 8 z + 12 a z + 17 a z + 6 8 9 3 9 5 9 2 10 4 10 > 13 a z + 8 a z + 17 a z + 9 a z + 3 a z + 3 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 248]], Vassiliev[3][Knot[11, Alternating, 248]]}
Out[15]=	{0, 0}
In[16]:=	Kh[Knot[11, Alternating, 248]][q, t]
Out[16]=	10 13 1 4 1 7 4 10 7 13 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 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 10 13 13 12 13 6 t 2 3 2 > ----- + ----- + ----- + ---- + ---- + --- + 9 q t + 3 q t + 6 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 + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a248

K11a247 K11a249