9 37

9_36

9_38

Visit 9 37's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 37's page at Knotilus! Visit 9 37's page at the original Knot Atlas!

9 37 Quick Notes

9 37 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X7,12,8,13 X3,11,4,10 X11,3,12,2 X5,14,6,15 X13,6,14,7 X15,18,16,1 X9,17,10,16 X17,9,18,8
Gauss code	-1, 4, -3, 1, -5, 6, -2, 9, -8, 3, -4, 2, -6, 5, -7, 8, -9, 7
Dowker-Thistlethwaite code	4 10 14 12 16 2 6 18 8
Conway Notation	[3,21,21]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 3 Super bridge index $\{4,7\}$ Nakanishi index 2 Maximal Thurston-Bennequin number [-6][-5] Hyperbolic Volume 10.9894 A-Polynomial See Data:9 37/A-polynomial

[edit Notes for 9 37's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $1$ Rasmussen s-Invariant 0

[edit Notes for 9 37's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{2}-11t+19-11t^{-1}+2t^{-2}$
Conway polynomial	$2z^{4}-3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{3,t+1\}$
Determinant and Signature	{ 45, 0 }
Jones polynomial	$q^{4}-2q^{3}+5q^{2}-7q+7-8q^{-1}+7q^{-2}-4q^{-3}+3q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{4}+z^{4}a^{2}+z^{2}a^{2}+2a^{2}+z^{4}-z^{2}-2-2z^{2}a^{-2}+a^{-4}$
Kauffman polynomial (db, data sources)	$a^{2}z^{8}+z^{8}+3a^{3}z^{7}+6az^{7}+3z^{7}a^{-1}+3a^{4}z^{6}+5a^{2}z^{6}+3z^{6}a^{-2}+5z^{6}+a^{5}z^{5}-6a^{3}z^{5}-13az^{5}-4z^{5}a^{-1}+2z^{5}a^{-3}-8a^{4}z^{4}-17a^{2}z^{4}-3z^{4}a^{-2}+z^{4}a^{-4}-13z^{4}-2a^{5}z^{3}+3a^{3}z^{3}+13az^{3}+6z^{3}a^{-1}-2z^{3}a^{-3}+5a^{4}z^{2}+14a^{2}z^{2}+z^{2}a^{-2}-2z^{2}a^{-4}+12z^{2}-2a^{3}z-7az-5za^{-1}-2a^{2}+a^{-4}-2$
The A2 invariant	$-q^{16}+q^{14}+q^{12}-q^{10}+3q^{8}+q^{6}-3-2q^{-4}+q^{-6}+2q^{-8}-q^{-10}+q^{-12}+q^{-14}$
The G2 invariant	$q^{80}-2q^{78}+4q^{76}-7q^{74}+5q^{72}-4q^{70}-4q^{68}+16q^{66}-23q^{64}+29q^{62}-22q^{60}+6q^{58}+16q^{56}-38q^{54}+52q^{52}-49q^{50}+24q^{48}+8q^{46}-33q^{44}+50q^{42}-44q^{40}+24q^{38}+7q^{36}-27q^{34}+33q^{32}-28q^{30}-6q^{28}+40q^{26}-46q^{24}+41q^{22}-18q^{20}-17q^{18}+57q^{16}-71q^{14}+65q^{12}-48q^{10}+4q^{8}+43q^{6}-65q^{4}+65q^{2}-47+16q^{-2}+18q^{-4}-39q^{-6}+32q^{-8}-22q^{-10}-7q^{-12}+33q^{-14}-38q^{-16}+22q^{-18}+6q^{-20}-33q^{-22}+50q^{-24}-48q^{-26}+29q^{-28}-8q^{-30}-20q^{-32}+38q^{-34}-40q^{-36}+34q^{-38}-14q^{-40}+2q^{-42}+9q^{-44}-15q^{-46}+15q^{-48}-12q^{-50}+8q^{-52}-2q^{-54}-q^{-56}+3q^{-58}-3q^{-60}+3q^{-62}-q^{-64}+q^{-66}$

Further Quantum Invariants

Further quantum knot invariants for 9_37.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+2q^{9}-q^{7}+3q^{5}-q^{3}-q-2q^{-3}+3q^{-5}-q^{-7}+q^{-9}$
2	$q^{32}-2q^{30}-2q^{28}+6q^{26}-2q^{24}-7q^{22}+10q^{20}+3q^{18}-12q^{16}+8q^{14}+6q^{12}-13q^{10}+6q^{6}-3q^{4}-4q^{2}+5+9q^{-2}-8q^{-4}-2q^{-6}+13q^{-8}-9q^{-10}-7q^{-12}+11q^{-14}-3q^{-16}-5q^{-18}+4q^{-20}-q^{-24}+q^{-26}$
3	$-q^{63}+2q^{61}+2q^{59}-3q^{57}-6q^{55}+2q^{53}+13q^{51}-q^{49}-19q^{47}-7q^{45}+26q^{43}+19q^{41}-25q^{39}-34q^{37}+22q^{35}+46q^{33}-8q^{31}-56q^{29}-5q^{27}+54q^{25}+15q^{23}-51q^{21}-30q^{19}+45q^{17}+32q^{15}-26q^{13}-33q^{11}+18q^{9}+34q^{7}+3q^{5}-32q^{3}-18q+26q^{-1}+32q^{-3}-20q^{-5}-51q^{-7}+10q^{-9}+56q^{-11}+4q^{-13}-58q^{-15}-11q^{-17}+49q^{-19}+24q^{-21}-38q^{-23}-25q^{-25}+25q^{-27}+21q^{-29}-10q^{-31}-16q^{-33}+4q^{-35}+8q^{-37}-2q^{-39}-4q^{-41}+q^{-43}+q^{-45}-q^{-49}+q^{-51}$
4	$q^{104}-2q^{102}-2q^{100}+3q^{98}+3q^{96}+6q^{94}-9q^{92}-13q^{90}+2q^{88}+10q^{86}+31q^{84}-8q^{82}-41q^{80}-26q^{78}+5q^{76}+82q^{74}+38q^{72}-48q^{70}-91q^{68}-69q^{66}+104q^{64}+137q^{62}+39q^{60}-122q^{58}-208q^{56}+11q^{54}+186q^{52}+197q^{50}-30q^{48}-287q^{46}-149q^{44}+110q^{42}+288q^{40}+120q^{38}-235q^{36}-241q^{34}-11q^{32}+262q^{30}+200q^{28}-122q^{26}-225q^{24}-87q^{22}+164q^{20}+187q^{18}-17q^{16}-167q^{14}-127q^{12}+61q^{10}+153q^{8}+85q^{6}-91q^{4}-160q^{2}-60+115q^{-2}+209q^{-4}+16q^{-6}-183q^{-8}-204q^{-10}+30q^{-12}+292q^{-14}+149q^{-16}-128q^{-18}-296q^{-20}-105q^{-22}+253q^{-24}+229q^{-26}+7q^{-28}-247q^{-30}-196q^{-32}+108q^{-34}+185q^{-36}+105q^{-38}-106q^{-40}-159q^{-42}-5q^{-44}+68q^{-46}+92q^{-48}-3q^{-50}-63q^{-52}-22q^{-54}+34q^{-58}+9q^{-60}-13q^{-62}-2q^{-64}-6q^{-66}+6q^{-68}+q^{-70}-3q^{-72}+2q^{-74}-2q^{-76}+q^{-78}-q^{-82}+q^{-84}$
5	$-q^{155}+2q^{153}+2q^{151}-3q^{149}-3q^{147}-3q^{145}+q^{143}+9q^{141}+13q^{139}-2q^{137}-20q^{135}-22q^{133}-7q^{131}+24q^{129}+49q^{127}+36q^{125}-30q^{123}-87q^{121}-77q^{119}+q^{117}+113q^{115}+163q^{113}+70q^{111}-122q^{109}-251q^{107}-195q^{105}+42q^{103}+322q^{101}+387q^{99}+126q^{97}-304q^{95}-569q^{93}-409q^{91}+149q^{89}+686q^{87}+733q^{85}+168q^{83}-652q^{81}-1042q^{79}-583q^{77}+432q^{75}+1201q^{73}+1037q^{71}-51q^{69}-1202q^{67}-1392q^{65}-396q^{63}+997q^{61}+1595q^{59}+828q^{57}-676q^{55}-1618q^{53}-1126q^{51}+330q^{49}+1462q^{47}+1286q^{45}+2q^{43}-1231q^{41}-1298q^{39}-210q^{37}+938q^{35}+1177q^{33}+372q^{31}-702q^{29}-1041q^{27}-424q^{25}+476q^{23}+875q^{21}+485q^{19}-293q^{17}-756q^{15}-533q^{13}+112q^{11}+662q^{9}+652q^{7}+95q^{5}-573q^{3}-803q-357q^{-1}+462q^{-3}+977q^{-5}+682q^{-7}-284q^{-9}-1125q^{-11}-1033q^{-13}+10q^{-15}+1166q^{-17}+1377q^{-19}+345q^{-21}-1089q^{-23}-1605q^{-25}-733q^{-27}+812q^{-29}+1696q^{-31}+1087q^{-33}-457q^{-35}-1549q^{-37}-1305q^{-39}+15q^{-41}+1246q^{-43}+1358q^{-45}+342q^{-47}-825q^{-49}-1191q^{-51}-576q^{-53}+399q^{-55}+907q^{-57}+640q^{-59}-75q^{-61}-579q^{-63}-543q^{-65}-127q^{-67}+285q^{-69}+390q^{-71}+185q^{-73}-99q^{-75}-223q^{-77}-155q^{-79}-2q^{-81}+105q^{-83}+96q^{-85}+29q^{-87}-36q^{-89}-49q^{-91}-20q^{-93}+9q^{-95}+16q^{-97}+11q^{-99}+4q^{-101}-8q^{-103}-5q^{-105}+2q^{-107}-q^{-109}+3q^{-113}-q^{-115}-2q^{-117}+q^{-119}-q^{-123}+q^{-125}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{16}+q^{14}+q^{12}-q^{10}+3q^{8}+q^{6}-3-2q^{-4}+q^{-6}+2q^{-8}-q^{-10}+q^{-12}+q^{-14}$
1,1	$q^{44}-4q^{42}+10q^{40}-22q^{38}+42q^{36}-70q^{34}+100q^{32}-136q^{30}+169q^{28}-186q^{26}+186q^{24}-156q^{22}+117q^{20}-40q^{18}-44q^{16}+138q^{14}-229q^{12}+284q^{10}-348q^{8}+352q^{6}-343q^{4}+300q^{2}-214+146q^{-2}-42q^{-4}-30q^{-6}+104q^{-8}-154q^{-10}+166q^{-12}-170q^{-14}+152q^{-16}-132q^{-18}+99q^{-20}-72q^{-22}+50q^{-24}-32q^{-26}+21q^{-28}-10q^{-30}+6q^{-32}-2q^{-34}+q^{-36}$
2,0	$q^{42}-q^{40}-2q^{38}+3q^{34}+q^{32}-6q^{30}+8q^{26}+5q^{24}-6q^{22}-2q^{20}+8q^{18}+4q^{16}-9q^{14}-5q^{12}+2q^{10}-5q^{8}-4q^{6}+2q^{2}+2+8q^{-2}+6q^{-4}-3q^{-6}-q^{-8}+8q^{-10}-11q^{-14}+7q^{-18}-8q^{-22}-3q^{-24}+4q^{-26}+2q^{-28}-q^{-30}+q^{-34}+q^{-36}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{34}-2q^{32}+2q^{28}-6q^{26}+4q^{24}+6q^{22}-6q^{20}+7q^{18}+8q^{16}-11q^{14}+2q^{10}-9q^{8}-2q^{6}+4q^{4}+6q^{2}+1+q^{-2}+8q^{-4}-5q^{-6}-10q^{-8}+8q^{-10}-4q^{-12}-8q^{-14}+9q^{-16}-3q^{-20}+4q^{-22}+q^{-24}-q^{-26}+q^{-28}$
1,0,0	$-q^{21}+q^{19}+q^{15}-q^{13}+3q^{11}+q^{9}+2q^{7}-2q-3q^{-1}-2q^{-5}+q^{-7}+2q^{-11}-q^{-13}+q^{-15}+q^{-17}+q^{-19}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{34}+2q^{32}-4q^{30}+6q^{28}-8q^{26}+10q^{24}-10q^{22}+10q^{20}-7q^{18}+6q^{16}+q^{14}-4q^{12}+12q^{10}-15q^{8}+18q^{6}-20q^{4}+18q^{2}-19+11q^{-2}-8q^{-4}+q^{-6}+2q^{-8}-6q^{-10}+10q^{-12}-10q^{-14}+11q^{-16}-8q^{-18}+7q^{-20}-4q^{-22}+3q^{-24}-q^{-26}+q^{-28}$
1,0	$q^{56}-2q^{52}-2q^{50}+2q^{48}+4q^{46}-2q^{44}-7q^{42}-2q^{40}+9q^{38}+8q^{36}-4q^{34}-9q^{32}+2q^{30}+12q^{28}+7q^{26}-9q^{24}-9q^{22}+2q^{20}+7q^{18}-4q^{16}-10q^{14}-2q^{12}+6q^{10}+2q^{8}-6q^{6}-q^{4}+8q^{2}+9-3q^{-2}-5q^{-4}+4q^{-6}+9q^{-8}-2q^{-10}-11q^{-12}-5q^{-14}+8q^{-16}+7q^{-18}-7q^{-20}-11q^{-22}+10q^{-26}+4q^{-28}-4q^{-30}-5q^{-32}+q^{-34}+4q^{-36}+2q^{-38}-q^{-40}-q^{-42}+q^{-46}$

G2 Invariants.

Weight

Invariant

1,0

$q^{80}-2q^{78}+4q^{76}-7q^{74}+5q^{72}-4q^{70}-4q^{68}+16q^{66}-23q^{64}+29q^{62}-22q^{60}+6q^{58}+16q^{56}-38q^{54}+52q^{52}-49q^{50}+24q^{48}+8q^{46}-33q^{44}+50q^{42}-44q^{40}+24q^{38}+7q^{36}-27q^{34}+33q^{32}-28q^{30}-6q^{28}+40q^{26}-46q^{24}+41q^{22}-18q^{20}-17q^{18}+57q^{16}-71q^{14}+65q^{12}-48q^{10}+4q^{8}+43q^{6}-65q^{4}+65q^{2}-47+16q^{-2}+18q^{-4}-39q^{-6}+32q^{-8}-22q^{-10}-7q^{-12}+33q^{-14}-38q^{-16}+22q^{-18}+6q^{-20}-33q^{-22}+50q^{-24}-48q^{-26}+29q^{-28}-8q^{-30}-20q^{-32}+38q^{-34}-40q^{-36}+34q^{-38}-14q^{-40}+2q^{-42}+9q^{-44}-15q^{-46}+15q^{-48}-12q^{-50}+8q^{-52}-2q^{-54}-q^{-56}+3q^{-58}-3q^{-60}+3q^{-62}-q^{-64}+q^{-66}$

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 37"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$2t^{2}-11t+19-11t^{-1}+2t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$2z^{4}-3z^{2}+1$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{3,t+1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 45, 0 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$q^{4}-2q^{3}+5q^{2}-7q+7-8q^{-1}+7q^{-2}-4q^{-3}+3q^{-4}-q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$-z^{2}a^{4}+z^{4}a^{2}+z^{2}a^{2}+2a^{2}+z^{4}-z^{2}-2-2z^{2}a^{-2}+a^{-4}$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$a^{2}z^{8}+z^{8}+3a^{3}z^{7}+6az^{7}+3z^{7}a^{-1}+3a^{4}z^{6}+5a^{2}z^{6}+3z^{6}a^{-2}+5z^{6}+a^{5}z^{5}-6a^{3}z^{5}-13az^{5}-4z^{5}a^{-1}+2z^{5}a^{-3}-8a^{4}z^{4}-17a^{2}z^{4}-3z^{4}a^{-2}+z^{4}a^{-4}-13z^{4}-2a^{5}z^{3}+3a^{3}z^{3}+13az^{3}+6z^{3}a^{-1}-2z^{3}a^{-3}+5a^{4}z^{2}+14a^{2}z^{2}+z^{2}a^{-2}-2z^{2}a^{-4}+12z^{2}-2a^{3}z-7az-5za^{-1}-2a^{2}+a^{-4}-2$

Vassiliev invariants

V2 and V3:

(-3, -1)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $-12$ $-8$ $72$ $82$ $22$ $96$ ${\frac {496}{3}}$ ${\frac {160}{3}}$ $24$ $-288$ $32$ $-984$ $-264$ $-{\frac {8191}{10}}$ $-{\frac {974}{15}}$ $-{\frac {1954}{5}}$ ${\frac {85}{2}}$ $-{\frac {671}{10}}$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). 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 9 37. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	-5	-4	-3	-2	-1	0	1	2	3	4	χ
9										1	1
7									1		-1
5								4	1		3
3							3	1			-2
1						4	4				0
-1					5	4					-1
-3				2	3						-1
-5			2	5							3
-7		1	2								-1
-9		2									2
-11	1										-1

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[9, 37]]
Out[2]=	9
In[3]:=	PD[Knot[9, 37]]
Out[3]=	PD[X[1, 4, 2, 5], X[7, 12, 8, 13], X[3, 11, 4, 10], X[11, 3, 12, 2], X[5, 14, 6, 15], X[13, 6, 14, 7], X[15, 18, 16, 1], X[9, 17, 10, 16], X[17, 9, 18, 8]]
In[4]:=	GaussCode[Knot[9, 37]]
Out[4]=	GaussCode[-1, 4, -3, 1, -5, 6, -2, 9, -8, 3, -4, 2, -6, 5, -7, 8, -9, 7]
In[5]:=	BR[Knot[9, 37]]
Out[5]=	BR[5, {-1, -1, 2, -1, -3, 2, 1, 4, -3, 2, -3, 4}]
In[6]:=	alex = Alexander[Knot[9, 37]][t]
Out[6]=	2 11 2 19 + -- - -- - 11 t + 2 t 2 t t
In[7]:=	Conway[Knot[9, 37]][z]
Out[7]=	2 4 1 - 3 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 37], Knot[11, NonAlternating, 100]}
In[9]:=	{KnotDet[Knot[9, 37]], KnotSignature[Knot[9, 37]]}
Out[9]=	{45, 0}
In[10]:=	J=Jones[Knot[9, 37]][q]
Out[10]=	-5 3 4 7 8 2 3 4 7 - q + -- - -- + -- - - - 7 q + 5 q - 2 q + q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 37]}
In[12]:=	A2Invariant[Knot[9, 37]][q]
Out[12]=	-16 -14 -12 -10 3 -6 4 6 8 10 -3 - q + q + q - q + -- + q - 2 q + q + 2 q - q + 8 q 12 14 q + q
In[13]:=	Kauffman[Knot[9, 37]][a, z]
Out[13]=	2 2 -4 2 5 z 3 2 2 z z 2 2 -2 + a - 2 a - --- - 7 a z - 2 a z + 12 z - ---- + -- + 14 a z + a 4 2 a a 3 3 4 4 2 2 z 6 z 3 3 3 5 3 4 z 5 a z - ---- + ---- + 13 a z + 3 a z - 2 a z - 13 z + -- - 3 a 4 a a 4 5 5 3 z 2 4 4 4 2 z 4 z 5 3 5 5 5 ---- - 17 a z - 8 a z + ---- - ---- - 13 a z - 6 a z + a z + 2 3 a a a 6 7 6 3 z 2 6 4 6 3 z 7 3 7 8 2 8 5 z + ---- + 5 a z + 3 a z + ---- + 6 a z + 3 a z + z + a z 2 a a
In[14]:=	{Vassiliev[2][Knot[9, 37]], Vassiliev[3][Knot[9, 37]]}
Out[14]=	{0, -1}
In[15]:=	Kh[Knot[9, 37]][q, t]
Out[15]=	4 1 2 1 2 2 5 2 - + 4 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 q t q t q t q t q t q t q t 3 5 3 3 2 5 2 5 3 7 3 9 4 ---- + --- + 4 q t + 3 q t + q t + 4 q t + q t + q t + q t 3 q t q t