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

K11a272 K11a274

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 10 14 16 20 4 18 2 22 12 8

Alexander Polynomial:

3t-3 - 16t-2 + 37t-1 - 47 + 37t - 16t2 + 3t3

Conway Polynomial:

1 + 2z4 + 3z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{159, -2}

Jones Polynomial:

q-9 - 4q-8 + 10q-7 - 17q-6 + 22q-5 - 26q-4 + 26q-3 - 22q-2 + 17q-1 - 9 + 4q - q2

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

{...}

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

- z2 - z4 + 2a2 + a2z2 + a2z4 + a2z6 + 4a4z2 + 5a4z4 + 2a4z6 - 2a6 - 5a6z2 - 3a6z4 + a8 + a8z2

Kauffman Polynomial:

- a-1z3 + a-1z5 + 2z2 - 5z4 + 4z6 + 4az3 - 10az5 + 8az7 - 2a2 - a2z2 + 10a2z4 - 16a2z6 + 11a2z8 + 3a3z + a3z3 - 3a3z5 - 7a3z7 + 9a3z9 - 13a4z2 + 40a4z4 - 47a4z6 + 17a4z8 + 3a4z10 + 5a5z - 8a5z3 + 15a5z5 - 30a5z7 + 17a5z9 + 2a6 - 16a6z2 + 39a6z4 - 45a6z6 + 14a6z8 + 3a6z10 + a7z + a7z3 - a7z5 - 11a7z7 + 8a7z9 + a8 - 5a8z2 + 12a8z4 - 17a8z6 + 8a8z8 - a9z + 5a9z3 - 8a9z5 + 4a9z7 + a10z2 - 2a10z4 + a10z6

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=-2 is the signature of 11273. 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                     3   j = 1                   6 1   j = -1                 11 3     j = -3               12 7       j = -5             14 10         j = -7           12 12           j = -9         10 14             j = -11       7 12               j = -13     3 10                 j = -15   1 7                   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, 273]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 273]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 3, 11, 4], X[14, 5, 15, 6], X[16, 8, 17, 7], > X[20, 9, 21, 10], X[4, 11, 5, 12], X[18, 13, 19, 14], X[2, 15, 3, 16], > X[22, 18, 1, 17], X[12, 19, 13, 20], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 273]]
Out[4]=	GaussCode[1, -8, 2, -6, 3, -1, 4, -11, 5, -2, 6, -10, 7, -3, 8, -4, 9, -7, 10, > -5, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 273]]
Out[5]=	DTCode[6, 10, 14, 16, 20, 4, 18, 2, 22, 12, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 273]][t]
Out[6]=	3 16 37 2 3 -47 + -- - -- + -- + 37 t - 16 t + 3 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 273]][z]
Out[7]=	4 6 1 + 2 z + 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 273]}
In[9]:=	{KnotDet[Knot[11, Alternating, 273]], KnotSignature[Knot[11, Alternating, 273]]}
Out[9]=	{159, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 273]][q]
Out[10]=	-9 4 10 17 22 26 26 22 17 2 -9 + q - -- + -- - -- + -- - -- + -- - -- + -- + 4 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, 273]}
In[12]:=	A2Invariant[Knot[11, Alternating, 273]][q]
Out[12]=	-28 2 4 3 -18 3 6 3 3 2 4 3 -2 + q - --- + --- - --- - q + --- - --- + --- - --- + -- + -- - -- + 24 22 20 16 14 12 10 8 6 4 q q q q q q q q q q 6 2 4 6 > -- - q + 2 q - q 2 q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 273]][a, z]
Out[13]=	2 6 8 2 2 2 4 2 6 2 8 2 4 2 4 2 a - 2 a + a - z + a z + 4 a z - 5 a z + a z - z + a z + 4 4 6 4 2 6 4 6 > 5 a z - 3 a z + a z + 2 a z
In[14]:=	Kauffman[Knot[11, Alternating, 273]][a, z]
Out[14]=	2 6 8 3 5 7 9 2 2 2 4 2 -2 a + 2 a + a + 3 a z + 5 a z + a z - a z + 2 z - a z - 13 a z - 3 6 2 8 2 10 2 z 3 3 3 5 3 7 3 > 16 a z - 5 a z + a z - -- + 4 a z + a z - 8 a z + a z + a 9 3 4 2 4 4 4 6 4 8 4 10 4 > 5 a z - 5 z + 10 a z + 40 a z + 39 a z + 12 a z - 2 a z + 5 z 5 3 5 5 5 7 5 9 5 6 2 6 > -- - 10 a z - 3 a z + 15 a z - a z - 8 a z + 4 z - 16 a z - a 4 6 6 6 8 6 10 6 7 3 7 5 7 > 47 a z - 45 a z - 17 a z + a z + 8 a z - 7 a z - 30 a z - 7 7 9 7 2 8 4 8 6 8 8 8 3 9 > 11 a z + 4 a z + 11 a z + 17 a z + 14 a z + 8 a z + 9 a z + 5 9 7 9 4 10 6 10 > 17 a z + 8 a z + 3 a z + 3 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 273]], Vassiliev[3][Knot[11, Alternating, 273]]}
Out[15]=	{0, 2}
In[16]:=	Kh[Knot[11, Alternating, 273]][q, t]
Out[16]=	7 11 1 3 1 7 3 10 7 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 12 14 10 12 3 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 > 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a273

K11a272 K11a274