The Alternating Knot 10₂₇

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Acknowledgement

KnotPlot

PD Presentation: X₄₂₅₁ X_12,4,13,3 X_20,13,1,14 X_14,5,15,6 X_6,19,7,20 X_18,9,19,10 X_16,7,17,8 X_8,17,9,18 X_10,15,11,16 X_2,12,3,11

Gauss Code: {1, -10, 2, -1, 4, -5, 7, -8, 6, -9, 10, -2, 3, -4, 9, -7, 8, -6, 5, -3}

DT (Dowker-Thistlethwaite) Code: 4 12 14 16 18 2 20 10 8 6

Minimum Braid Representative:

Length is 11, width is 4
Braid index is 4

A Morse Link Presentation:

3D Invariants:

Symmetry Type Unknotting Number 3-Genus Bridge/Super Bridge Index Nakanishi Index

Reversible 1 3 2 / NotAvailable 1

Alexander Polynomial: 2t^-3 - 8t^-2 + 16t^-1 - 19 + 16t - 8t² + 2t³

Conway Polynomial: 1 + 2z² + 4z⁴ + 2z⁶

Other knots with the same Alexander/Conway Polynomial: {...}

Determinant and Signature: {71, -2}

Jones Polynomial: - q^-8 + 3q^-7 - 6q^-6 + 9q^-5 - 11q^-4 + 12q^-3 - 11q^-2 + 9q^-1 - 5 + 3q - q²

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

A2 (sl(3)) Invariant: - q^-24 + q^-22 - q^-20 - q^-18 + 2q^-16 - 2q^-14 + 2q^-12 + 2q^-6 - 2q^-4 + 3q^-2 + q⁴ - q⁶

HOMFLY-PT Polynomial: - 2z² - z⁴ + a² + 3a²z² + 3a²z⁴ + a²z⁶ + a⁴ + 3a⁴z² + 3a⁴z⁴ + a⁴z⁶ - a⁶ - 2a⁶z² - a⁶z⁴

Kauffman Polynomial: - 2a^-1z³ + a^-1z⁵ + 4z² - 7z⁴ + 3z⁶ - az + 5az³ - 8az⁵ + 4az⁷ - a² + 4a²z² - 3a²z⁴ - 2a²z⁶ + 3a²z⁸ - 2a³z + 11a³z³ - 14a³z⁵ + 6a³z⁷ + a³z⁹ + a⁴ - 4a⁴z² + 7a⁴z⁴ - 9a⁴z⁶ + 6a⁴z⁸ - 2a⁵z + 7a⁵z³ - 12a⁵z⁵ + 6a⁵z⁷ + a⁵z⁹ + a⁶ - a⁶z² - 3a⁶z⁴ - a⁶z⁶ + 3a⁶z⁸ + a⁷z³ - 6a⁷z⁵ + 4a⁷z⁷ + 3a⁸z² - 6a⁸z⁴ + 3a⁸z⁶ + a⁹z - 2a⁹z³ + 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 10₂₇. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

t^rq^j 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 2

j = 1 3 1

j = -1 6 2

j = -3 6 4

j = -5 6 5

j = -7 5 6

j = -9 4 6

j = -11 2 5

j = -13 1 4

j = -15 2

j = -17 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-23 - 3q^-22 + q^-21 + 9q^-20 - 16q^-19 - q^-18 + 35q^-17 - 39q^-16 - 15q^-15 + 77q^-14 - 60q^-13 - 41q^-12 + 116q^-11 - 66q^-10 - 66q^-9 + 131q^-8 - 54q^-7 - 75q^-6 + 112q^-5 - 30q^-4 - 63q^-3 + 70q^-2 - 8q^-1 - 37 + 30q + q² - 15q³ + 8q⁴ + q⁵ - 3q⁶ + q⁷

3 - q^-45 + 3q^-44 - q^-43 - 4q^-42 - 2q^-41 + 13q^-40 + 4q^-39 - 26q^-38 - 13q^-37 + 47q^-36 + 31q^-35 - 72q^-34 - 68q^-33 + 105q^-32 + 119q^-31 - 130q^-30 - 192q^-29 + 145q^-28 + 281q^-27 - 147q^-26 - 374q^-25 + 128q^-24 + 467q^-23 - 95q^-22 - 545q^-21 + 47q^-20 + 607q^-19 + 3q^-18 - 639q^-17 - 59q^-16 + 647q^-15 + 107q^-14 - 618q^-13 - 158q^-12 + 571q^-11 + 188q^-10 - 490q^-9 - 217q^-8 + 406q^-7 + 214q^-6 - 297q^-5 - 216q^-4 + 219q^-3 + 177q^-2 - 129q^-1 - 150 + 79q + 105q² - 35q³ - 73q⁴ + 16q⁵ + 43q⁶ - 5q⁷ - 24q⁸ + q⁹ + 12q¹⁰ - q¹¹ - 4q¹² - q¹³ + 3q¹⁴ - q¹⁵

4 q^-74 - 3q^-73 + q^-72 + 4q^-71 - 3q^-70 + 5q^-69 - 16q^-68 + 4q^-67 + 22q^-66 - 9q^-65 + 16q^-64 - 66q^-63 + q^-62 + 83q^-61 + 7q^-60 + 47q^-59 - 209q^-58 - 58q^-57 + 200q^-56 + 123q^-55 + 184q^-54 - 491q^-53 - 304q^-52 + 284q^-51 + 409q^-50 + 607q^-49 - 810q^-48 - 838q^-47 + 103q^-46 + 764q^-45 + 1443q^-44 - 906q^-43 - 1549q^-42 - 495q^-41 + 935q^-40 + 2535q^-39 - 607q^-38 - 2136q^-37 - 1381q^-36 + 761q^-35 + 3532q^-34 - 17q^-33 - 2385q^-32 - 2233q^-31 + 323q^-30 + 4140q^-29 + 620q^-28 - 2266q^-27 - 2804q^-26 - 218q^-25 + 4250q^-24 + 1141q^-23 - 1838q^-22 - 2991q^-21 - 764q^-20 + 3843q^-19 + 1478q^-18 - 1154q^-17 - 2766q^-16 - 1234q^-15 + 2978q^-14 + 1549q^-13 - 357q^-12 - 2140q^-11 - 1471q^-10 + 1853q^-9 + 1289q^-8 + 287q^-7 - 1291q^-6 - 1340q^-5 + 846q^-4 + 787q^-3 + 539q^-2 - 536q^-1 - 913 + 244q + 306q² + 440q³ - 112q⁴ - 462q⁵ + 36q⁶ + 44q⁷ + 228q⁸ + 16q⁹ - 178q¹⁰ + 8q¹¹ - 21q¹² + 82q¹³ + 19q¹⁴ - 56q¹⁵ + 9q¹⁶ - 14q¹⁷ + 21q¹⁸ + 7q¹⁹ - 15q²⁰ + 4q²¹ - 3q²² + 4q²³ + q²⁴ - 3q²⁵ + q²⁶

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]:=
PD[Knot[10, 27]]

Out[2]=
PD[X[4, 2, 5, 1], X[12, 4, 13, 3], X[20, 13, 1, 14], X[14, 5, 15, 6], > X[6, 19, 7, 20], X[18, 9, 19, 10], X[16, 7, 17, 8], X[8, 17, 9, 18], > X[10, 15, 11, 16], X[2, 12, 3, 11]]

In[3]:=
GaussCode[Knot[10, 27]]

Out[3]=
GaussCode[1, -10, 2, -1, 4, -5, 7, -8, 6, -9, 10, -2, 3, -4, 9, -7, 8, -6, 5, > -3]

In[4]:=
DTCode[Knot[10, 27]]

Out[4]=
DTCode[4, 12, 14, 16, 18, 2, 20, 10, 8, 6]

In[5]:=
br = BR[Knot[10, 27]]

Out[5]=
BR[4, {-1, -1, -1, -1, -2, 1, -2, 3, -2, 3, 3}]

In[6]:=
{First[br], Crossings[br]}

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[10, 27]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[10, 27]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 27]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

Out[9]=
{Reversible, 1, 3, 2, NotAvailable, 1}

In[10]:=
alex = Alexander[Knot[10, 27]][t]

Out[10]=
2 8 16 2 3 -19 + -- - -- + -- + 16 t - 8 t + 2 t 3 2 t t t

In[11]:=
Conway[Knot[10, 27]][z]

Out[11]=
2 4 6 1 + 2 z + 4 z + 2 z

In[12]:=
Select[AllKnots[], (alex === Alexander[#][t])&]

Out[12]=
{Knot[10, 27]}

In[13]:=
{KnotDet[Knot[10, 27]], KnotSignature[Knot[10, 27]]}

Out[13]=
{71, -2}

In[14]:=
Jones[Knot[10, 27]][q]

Out[14]=
-8 3 6 9 11 12 11 9 2 -5 - q + -- - -- + -- - -- + -- - -- + - + 3 q - q 7 6 5 4 3 2 q q q q q q q

In[15]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]

Out[15]=
{Knot[10, 27]}

In[16]:=
A2Invariant[Knot[10, 27]][q]

Out[16]=
-24 -22 -20 -18 2 2 2 2 2 3 4 6 -q + q - q - q + --- - --- + --- + -- - -- + -- + q - q 16 14 12 6 4 2 q q q q q q

In[17]:=
HOMFLYPT[Knot[10, 27]][a, z]

Out[17]=
2 4 6 2 2 2 4 2 6 2 4 2 4 4 4 a + a - a - 2 z + 3 a z + 3 a z - 2 a z - z + 3 a z + 3 a z - 6 4 2 6 4 6 > a z + a z + a z

In[18]:=
Kauffman[Knot[10, 27]][a, z]

Out[18]=
2 4 6 3 5 9 2 2 2 4 2 -a + a + a - a z - 2 a z - 2 a z + a z + 4 z + 4 a z - 4 a z - 3 6 2 8 2 2 z 3 3 3 5 3 7 3 9 3 > a z + 3 a z - ---- + 5 a z + 11 a z + 7 a z + a z - 2 a z - a 5 4 2 4 4 4 6 4 8 4 z 5 3 5 > 7 z - 3 a z + 7 a z - 3 a z - 6 a z + -- - 8 a z - 14 a z - a 5 5 7 5 9 5 6 2 6 4 6 6 6 8 6 > 12 a z - 6 a z + a z + 3 z - 2 a z - 9 a z - a z + 3 a z + 7 3 7 5 7 7 7 2 8 4 8 6 8 > 4 a z + 6 a z + 6 a z + 4 a z + 3 a z + 6 a z + 3 a z + 3 9 5 9 > a z + a z

In[19]:=
{Vassiliev[2][Knot[10, 27]], Vassiliev[3][Knot[10, 27]]}

Out[19]=
{2, -3}

In[20]:=
Kh[Knot[10, 27]][q, t]

Out[20]=
4 6 1 2 1 4 2 5 4 6 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 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 5 6 6 5 6 2 t 2 3 2 5 3 > ----- + ----- + ----- + ---- + ---- + --- + 3 q t + q t + 2 q t + q t 7 3 7 2 5 2 5 3 q q t q t q t q t q t

In[21]:=
ColouredJones[Knot[10, 27], 2][q]

Out[21]=
-23 3 -21 9 16 -18 35 39 15 77 60 -37 + q - --- + q + --- - --- - q + --- - --- - --- + --- - --- - 22 20 19 17 16 15 14 13 q q q q q q q q 41 116 66 66 131 54 75 112 30 63 70 8 2 > --- + --- - --- - -- + --- - -- - -- + --- - -- - -- + -- - - + 30 q + q - 12 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q q 3 4 5 6 7 > 15 q + 8 q + q - 3 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₂₇

10₂₆
10₂₈

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-23 - 3q^-22 + q^-21 + 9q^-20 - 16q^-19 - q^-18 + 35q^-17 - 39q^-16 - 15q^-15 + 77q^-14 - 60q^-13 - 41q^-12 + 116q^-11 - 66q^-10 - 66q^-9 + 131q^-8 - 54q^-7 - 75q^-6 + 112q^-5 - 30q^-4 - 63q^-3 + 70q^-2 - 8q^-1 - 37 + 30q + q² - 15q³ + 8q⁴ + q⁵ - 3q⁶ + q⁷
3	- q^-45 + 3q^-44 - q^-43 - 4q^-42 - 2q^-41 + 13q^-40 + 4q^-39 - 26q^-38 - 13q^-37 + 47q^-36 + 31q^-35 - 72q^-34 - 68q^-33 + 105q^-32 + 119q^-31 - 130q^-30 - 192q^-29 + 145q^-28 + 281q^-27 - 147q^-26 - 374q^-25 + 128q^-24 + 467q^-23 - 95q^-22 - 545q^-21 + 47q^-20 + 607q^-19 + 3q^-18 - 639q^-17 - 59q^-16 + 647q^-15 + 107q^-14 - 618q^-13 - 158q^-12 + 571q^-11 + 188q^-10 - 490q^-9 - 217q^-8 + 406q^-7 + 214q^-6 - 297q^-5 - 216q^-4 + 219q^-3 + 177q^-2 - 129q^-1 - 150 + 79q + 105q² - 35q³ - 73q⁴ + 16q⁵ + 43q⁶ - 5q⁷ - 24q⁸ + q⁹ + 12q¹⁰ - q¹¹ - 4q¹² - q¹³ + 3q¹⁴ - q¹⁵
4	q^-74 - 3q^-73 + q^-72 + 4q^-71 - 3q^-70 + 5q^-69 - 16q^-68 + 4q^-67 + 22q^-66 - 9q^-65 + 16q^-64 - 66q^-63 + q^-62 + 83q^-61 + 7q^-60 + 47q^-59 - 209q^-58 - 58q^-57 + 200q^-56 + 123q^-55 + 184q^-54 - 491q^-53 - 304q^-52 + 284q^-51 + 409q^-50 + 607q^-49 - 810q^-48 - 838q^-47 + 103q^-46 + 764q^-45 + 1443q^-44 - 906q^-43 - 1549q^-42 - 495q^-41 + 935q^-40 + 2535q^-39 - 607q^-38 - 2136q^-37 - 1381q^-36 + 761q^-35 + 3532q^-34 - 17q^-33 - 2385q^-32 - 2233q^-31 + 323q^-30 + 4140q^-29 + 620q^-28 - 2266q^-27 - 2804q^-26 - 218q^-25 + 4250q^-24 + 1141q^-23 - 1838q^-22 - 2991q^-21 - 764q^-20 + 3843q^-19 + 1478q^-18 - 1154q^-17 - 2766q^-16 - 1234q^-15 + 2978q^-14 + 1549q^-13 - 357q^-12 - 2140q^-11 - 1471q^-10 + 1853q^-9 + 1289q^-8 + 287q^-7 - 1291q^-6 - 1340q^-5 + 846q^-4 + 787q^-3 + 539q^-2 - 536q^-1 - 913 + 244q + 306q² + 440q³ - 112q⁴ - 462q⁵ + 36q⁶ + 44q⁷ + 228q⁸ + 16q⁹ - 178q¹⁰ + 8q¹¹ - 21q¹² + 82q¹³ + 19q¹⁴ - 56q¹⁵ + 9q¹⁶ - 14q¹⁷ + 21q¹⁸ + 7q¹⁹ - 15q²⁰ + 4q²¹ - 3q²² + 4q²³ + q²⁴ - 3q²⁵ + q²⁶

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 27]]
Out[2]=	PD[X[4, 2, 5, 1], X[12, 4, 13, 3], X[20, 13, 1, 14], X[14, 5, 15, 6], > X[6, 19, 7, 20], X[18, 9, 19, 10], X[16, 7, 17, 8], X[8, 17, 9, 18], > X[10, 15, 11, 16], X[2, 12, 3, 11]]
In[3]:=	GaussCode[Knot[10, 27]]
Out[3]=	GaussCode[1, -10, 2, -1, 4, -5, 7, -8, 6, -9, 10, -2, 3, -4, 9, -7, 8, -6, 5, > -3]
In[4]:=	DTCode[Knot[10, 27]]
Out[4]=	DTCode[4, 12, 14, 16, 18, 2, 20, 10, 8, 6]
In[5]:=	br = BR[Knot[10, 27]]
Out[5]=	BR[4, {-1, -1, -1, -1, -2, 1, -2, 3, -2, 3, 3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 27]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 27]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 27]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 3, 2, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 27]][t]
Out[10]=	2 8 16 2 3 -19 + -- - -- + -- + 16 t - 8 t + 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 27]][z]
Out[11]=	2 4 6 1 + 2 z + 4 z + 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 27]}
In[13]:=	{KnotDet[Knot[10, 27]], KnotSignature[Knot[10, 27]]}
Out[13]=	{71, -2}
In[14]:=	Jones[Knot[10, 27]][q]
Out[14]=	-8 3 6 9 11 12 11 9 2 -5 - q + -- - -- + -- - -- + -- - -- + - + 3 q - q 7 6 5 4 3 2 q q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 27]}
In[16]:=	A2Invariant[Knot[10, 27]][q]
Out[16]=	-24 -22 -20 -18 2 2 2 2 2 3 4 6 -q + q - q - q + --- - --- + --- + -- - -- + -- + q - q 16 14 12 6 4 2 q q q q q q
In[17]:=	HOMFLYPT[Knot[10, 27]][a, z]
Out[17]=	2 4 6 2 2 2 4 2 6 2 4 2 4 4 4 a + a - a - 2 z + 3 a z + 3 a z - 2 a z - z + 3 a z + 3 a z - 6 4 2 6 4 6 > a z + a z + a z
In[18]:=	Kauffman[Knot[10, 27]][a, z]
Out[18]=	2 4 6 3 5 9 2 2 2 4 2 -a + a + a - a z - 2 a z - 2 a z + a z + 4 z + 4 a z - 4 a z - 3 6 2 8 2 2 z 3 3 3 5 3 7 3 9 3 > a z + 3 a z - ---- + 5 a z + 11 a z + 7 a z + a z - 2 a z - a 5 4 2 4 4 4 6 4 8 4 z 5 3 5 > 7 z - 3 a z + 7 a z - 3 a z - 6 a z + -- - 8 a z - 14 a z - a 5 5 7 5 9 5 6 2 6 4 6 6 6 8 6 > 12 a z - 6 a z + a z + 3 z - 2 a z - 9 a z - a z + 3 a z + 7 3 7 5 7 7 7 2 8 4 8 6 8 > 4 a z + 6 a z + 6 a z + 4 a z + 3 a z + 6 a z + 3 a z + 3 9 5 9 > a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 27]], Vassiliev[3][Knot[10, 27]]}
Out[19]=	{2, -3}
In[20]:=	Kh[Knot[10, 27]][q, t]
Out[20]=	4 6 1 2 1 4 2 5 4 6 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 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 5 6 6 5 6 2 t 2 3 2 5 3 > ----- + ----- + ----- + ---- + ---- + --- + 3 q t + q t + 2 q t + q t 7 3 7 2 5 2 5 3 q q t q t q t q t q t
In[21]:=	ColouredJones[Knot[10, 27], 2][q]
Out[21]=	-23 3 -21 9 16 -18 35 39 15 77 60 -37 + q - --- + q + --- - --- - q + --- - --- - --- + --- - --- - 22 20 19 17 16 15 14 13 q q q q q q q q 41 116 66 66 131 54 75 112 30 63 70 8 2 > --- + --- - --- - -- + --- - -- - -- + --- - -- - -- + -- - - + 30 q + q - 12 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q q 3 4 5 6 7 > 15 q + 8 q + q - 3 q + q

The Alternating Knot 1027

The Alternating Knot 10₂₇